{"id":209,"date":"2019-08-20T17:04:07","date_gmt":"2019-08-20T21:04:07","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/counting-principles\/"},"modified":"2022-06-01T10:39:40","modified_gmt":"2022-06-01T14:39:40","slug":"counting-principles","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/counting-principles\/","title":{"raw":"Counting Principles","rendered":"Counting Principles"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Solve counting problems using the Addition Principle.<\/li>\n \t<li>Solve counting problems using the Multiplication Principle.<\/li>\n \t<li>Solve counting problems using permutations involving n distinct objects.<\/li>\n \t<li>Solve counting problems using combinations.<\/li>\n \t<li>Find the number of subsets of a given set.<\/li>\n \t<li>Solve counting problems using permutations involving n non-distinct objects.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137432308\">A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!<\/p>\n<p id=\"fs-id1165135313702\">We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.<\/p>\n\n<div id=\"fs-id1165137463810\" class=\"bc-section section\">\n<h3>Using the Addition Principle<\/h3>\n<p id=\"fs-id1165137832066\">The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The <strong>Addition Principle<\/strong> tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options, as we can see in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_05_001\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_11_05_001\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154936\/CNX_Precalc_Figure_11_05_001n.jpg\" alt=\"The addition of 3 iPods and 4 iPhones.\" width=\"487\" height=\"358\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135186420\" class=\"textbox key-takeaways\">\n<h3>The Addition Principle<\/h3>\n<p id=\"fs-id1165135351541\">According to the Addition Principle, if one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first <em>or<\/em> second event can occur in [latex]m+n[\/latex] ways.<\/p>\n\n<\/div>\n<div id=\"Example_11_05_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137536266\">\n<div id=\"fs-id1165137552601\">\n<h3>Using the Addition Principle<\/h3>\n<p id=\"fs-id1165135206132\">There are 2 vegetarian entr\u00e9e options and 5 meat entr\u00e9e options on a dinner menu. What is the total number of entr\u00e9e options?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137410849\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137410849\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137410849\"]\n<p id=\"fs-id1165137399341\">We can add the number of vegetarian options to the number of meat options to find the total number of entr\u00e9e options.<\/p>\n<span id=\"fs-id1165135237185\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154938\/CNX_Precalc_Figure_11_05_002.jpg\" alt=\"The addition of the type of options for an entree.\"><\/span>\n<p id=\"fs-id1165137786497\">There are 7 total options.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843834\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_01\">\n<div id=\"fs-id1165137455516\">\n<p id=\"fs-id1165137529243\">A student is shopping for a new computer. He is deciding among 3 desktop computers and 4 laptop computers. What is the total number of computer options?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137723942\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137723942\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137723942\"]\n<p id=\"fs-id1165137732616\">7<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137645260\" class=\"bc-section section\">\n<h3>Using the Multiplication Principle<\/h3>\n<p id=\"fs-id1165137767730\">The <strong>Multiplication Principle<\/strong> applies when we are making more than one selection. Suppose we are choosing an appetizer, an entr\u00e9e, and a dessert. If there are 2 appetizer options, 3 entr\u00e9e options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in <a class=\"autogenerated-content\" href=\"#CNX_Precalculus_Figure_11_05_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalculus_Figure_11_05_003\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154940\/CNX_Precalc_Figure_11_05_003.jpg\" alt=\"A tree diagram of the different menu combinations.\" width=\"975\" height=\"287\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\nThe possible choices are:\n<ol id=\"fs-id1165137641640\" type=\"1\">\n \t<li>soup, chicken, cake<\/li>\n \t<li>soup, chicken, pudding<\/li>\n \t<li>soup, fish, cake<\/li>\n \t<li>soup, fish, pudding<\/li>\n \t<li>soup, steak, cake<\/li>\n \t<li>soup, steak, pudding<\/li>\n \t<li>salad, chicken, cake<\/li>\n \t<li>salad, chicken, pudding<\/li>\n \t<li>salad, fish, cake<\/li>\n \t<li>salad, fish, pudding<\/li>\n \t<li>salad, steak, cake<\/li>\n \t<li>salad, steak, pudding<\/li>\n<\/ol>\nWe can also find the total number of possible dinners by multiplying.\n<p id=\"eip-933\">We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.<\/p>\n\n<div id=\"eip-828\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{llll}\\#\\text{ of appetizer options }\u00d7\\hfill &amp; \\#\\text{ of entree options }\u00d7\\hfill &amp; \\#\\text{ of dessert options}\\hfill &amp; \\hfill \\\\ \\text{ }2\\text{ }\u00d7\\hfill &amp; \\text{ }3\\text{ }\u00d7\\hfill &amp; \\text{ }2\\hfill &amp; =12\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165137560649\" class=\"textbox key-takeaways\">\n<h3>The Multiplication Principle<\/h3>\n<p id=\"fs-id1165137706934\">According to the Multiplication Principle, if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways. This is also known as the Fundamental Counting Principle.<\/p>\n\n<\/div>\n<div id=\"Example_11_05_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137771642\">\n<div id=\"fs-id1165134047731\">\n<h3>Using the Multiplication Principle<\/h3>\n<p id=\"fs-id1165137590306\">Diane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137580422\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137580422\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137580422\"]\n<p id=\"fs-id1165137771530\">To find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and the number of sweater options.<\/p>\n<span id=\"fs-id1165137733725\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154946\/CNX_Precalc_Figure_11_05_004.jpg\" alt=\"The multiplication of number of skirt options (2) times the number of blouse options (4) times the number of sweater options (2) which equals 16.\"><\/span>\n<p id=\"fs-id1165137755612\">There are 16 possible outfits.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137398102\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_02\">\n<div id=\"fs-id1165137571190\">\n<p id=\"fs-id1165137423732\">A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137473501\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137473501\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137473501\"]\n<p id=\"fs-id1165135255245\">There are 60 possible breakfast specials.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137553058\" class=\"bc-section section\">\n<h3>Finding the Number of Permutations of <em>n<\/em> Distinct Objects<\/h3>\n<p id=\"fs-id1165137461547\">The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.<\/p>\n\n<div id=\"fs-id1165137678383\" class=\"bc-section section\">\n<h4>Finding the Number of Permutations of <em>n<\/em> Distinct Objects Using the Multiplication Principle<\/h4>\n<p id=\"fs-id1165137725216\">To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.<\/p>\n<span id=\"fs-id1165137804112\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155013\/CNX_Precalc_Figure_11_05_005.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137732561\">There are four options for the first place, so we write a 4 on the first line.<\/p>\n<span id=\"fs-id1165137854788\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155021\/CNX_Precalc_Figure_11_05_006.jpg\" alt=\"Four times two blanks spots.\"><\/span>\n<p id=\"fs-id1165137597142\">After the first place has been filled, there are three options for the second place so we write a 3 on the second line.<\/p>\n<span id=\"fs-id1165137570729\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155023\/CNX_Precalc_Figure_11_05_007.jpg\" alt=\"Four times three times one blank spot.\"><\/span>\n<p id=\"fs-id1165137529516\">After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.<\/p>\n<span id=\"eip-id1165137874239\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155036\/CNX_Precalc_Figure_11_05_008.jpg\" alt=\"\"><\/span>\n\nThere are 24 possible permutations of the paintings.\n<div id=\"fs-id1165137731358\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137436230\"><strong>Given<\/strong>[latex]\\,n\\,[\/latex]<strong>distinct options, determine how many permutations there are.<\/strong><\/p>\n\n<ol id=\"fs-id1165137552251\" type=\"1\">\n \t<li>Determine how many options there are for the first situation.<\/li>\n \t<li>Determine how many options are left for the second situation.<\/li>\n \t<li>Continue until all of the spots are filled.<\/li>\n \t<li>Multiply the numbers together.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_05_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137724829\">\n<div id=\"fs-id1165137551779\">\n<h3>Finding the Number of Permutations Using the Multiplication Principle<\/h3>\nAt a swimming competition, nine swimmers compete in a race.\n<ol id=\"fs-id1165137803400\" type=\"a\">\n \t<li>How many ways can they place first, second, and third?<\/li>\n \t<li>How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)<\/li>\n \t<li>How many ways can all nine swimmers line up for a photo?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137533114\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"797441\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"797441\"]\n<ol id=\"fs-id1165135177686\" type=\"a\">\n \t<li>Draw lines for each place.\n<span id=\"eip-id1165133349418\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155038\/CNX_Precalc_Figure_11_05_009.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137827584\">There are 9 options for first place. Once someone has won first place, there are 8 remaining options for second place. Once first and second place have been won, there are 7 remaining options for third place.<\/p>\n<span id=\"eip-id1165135317102\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155040\/CNX_Precalc_Figure_11_05_010.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137533708\">Multiply to find that there are 504 ways for the swimmers to place.<\/p>\n<\/li>\n \t<li>Draw lines for describing each place.\n<span id=\"eip-id1165133101712\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155042\/CNX_Precalc_Figure_11_05_011.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137527991\">We know Ariel must win first place, so there is only 1 option for first place. There are 8 remaining options for second place, and then 7 remaining options for third place.<\/p>\n<span id=\"eip-id1165137580736\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155049\/CNX_Precalc_Figure_11_05_012.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137464178\">Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first.<\/p>\n<\/li>\n \t<li>\n<p id=\"fs-id1165137531490\">Draw lines for describing each place in the photo.<\/p>\n<span id=\"eip-id1165135192171\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155051\/CNX_Precalc_Figure_11_05_013.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137779101\">There are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person remains for the last spot.<\/p>\n<span id=\"eip-id1165137551358\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155053\/CNX_Precalc_Figure_11_05_014.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137446031\">There are 362,880 possible permutations for the swimmers to line up.<\/p>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165137446031\">[\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137896132\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135638483\">Note that in part c, we found there were 9! ways for 9 people to line up. The number of permutations of[latex]\\,n\\,[\/latex]distinct objects can always be found by[latex]\\,n!.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-327\">A family of five is having portraits taken. Use the Multiplication Principle to find the following.<\/p>\n\n<div id=\"eip-id2465210\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_03\">\n<div id=\"fs-id1165137817722\">\n<p id=\"fs-id1165137817723\">How many ways can the family line up for the portrait?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137407672\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137407672\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137407672\"]\n<p id=\"fs-id1165137407673\">120<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id2465241\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_04\">\n<div id=\"fs-id1165135193101\">\n<p id=\"fs-id1165135193102\">How many ways can the photographer line up 3 family members?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137665821\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137665821\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137665821\"]\n<p id=\"fs-id1165137665822\">60<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id2465272\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_05\">\n<div id=\"fs-id1165137810891\">\n<p id=\"fs-id1165137810892\">How many ways can the family line up for the portrait if the parents are required to stand on each end?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137424182\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137424182\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137424182\"]\n<p id=\"fs-id1165137424183\">12<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137698292\" class=\"bc-section section\">\n<h4>Finding the Number of Permutations of <em>n<\/em> Distinct Objects Using a Formula<\/h4>\n<p id=\"fs-id1165137862888\">For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let\u2019s look at two common notations for permutations. If we have a set of[latex]\\,n\\,[\/latex]objects and we want to choose[latex]\\,r\\,[\/latex]objects from the set in order, we write[latex]\\,P\\left(n,r\\right).\\,[\/latex]Another way to write this is [latex]{n}_{}{P}_{r},\\,[\/latex]a notation commonly seen on computers and calculators. To calculate[latex]\\,P\\left(n,r\\right),\\,[\/latex]we begin by finding[latex]\\,n!,\\,[\/latex]the number of ways to line up all [latex]n[\/latex] objects. We then divide by[latex]\\,\\left(n-r\\right)!\\,[\/latex] to cancel out the[latex]\\,\\left(n-r\\right)\\,[\/latex]items that we do not wish to line up.<\/p>\n<p id=\"fs-id1165137530985\">Let\u2019s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is [latex]6\u00d75\u00d74=120.[\/latex] Using factorials, we get the same result.<\/p>\n\n<div id=\"eip-959\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{6!}{3!}=\\frac{6\u00b75\u00b74\u00b73!}{3!}=6\u00b75\u00b74=120\\,[\/latex]<\/div>\n<p id=\"fs-id1165137406160\">There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.<\/p>\n\n<div id=\"eip-469\" class=\"unnumbered aligncenter\">[latex]\\,P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137443996\">Note that the formula stills works if we are choosing <u>all<\/u>[latex]\\,n\\,[\/latex]objects and placing them in order. In that case we would be dividing by[latex]\\,\\left(n-n\\right)!\\,[\/latex]or[latex]\\,0!,\\,[\/latex]which we said earlier is equal to 1. So the number of permutations of[latex]\\,n\\,[\/latex]objects taken[latex]\\,n\\,[\/latex]at a time is[latex]\\,\\frac{n!}{1}\\,[\/latex]or just[latex]\\,n!\\text{.}[\/latex]<\/p>\n\n<div id=\"fs-id1165137657370\" class=\"textbox key-takeaways\">\n<h3>Formula for Permutations of <em>n<\/em> Distinct Objects<\/h3>\n<p id=\"fs-id1165135503766\">Given[latex]\\,n\\,[\/latex]distinct objects, the number of ways to select[latex]\\,r\\,[\/latex]objects from the set in order is<\/p>\n\n<div>[latex]P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137644544\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137531306\"><strong>Given a word problem, evaluate the possible permutations.<\/strong><\/p>\n\n<ol id=\"fs-id1165137461265\" type=\"1\">\n \t<li>Identify[latex]\\,n\\,[\/latex]from the given information.<\/li>\n \t<li>Identify[latex]\\,r\\,[\/latex]from the given information.<\/li>\n \t<li>Replace[latex]\\,n\\,[\/latex]and[latex]\\,r\\,[\/latex]in the formula with the given values.<\/li>\n \t<li>Evaluate.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137660582\">\n<div id=\"fs-id1165137660584\">\n<h3>Finding the Number of Permutations Using the Formula<\/h3>\n<p id=\"fs-id1165137677568\">A professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and arrange the questions?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137597587\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137597587\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137597587\"]\n<p id=\"fs-id1165137597589\">Substitute[latex]\\,n=12\\,[\/latex]and[latex]\\,r=9\\,[\/latex]into the permutation formula and simplify.<\/p>\n\n<div id=\"eip-id1165134385701\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{ }P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}\\,\\hfill \\\\ P\\left(12,9\\right)=\\frac{12!}{\\left(12-9\\right)!}=\\frac{12!}{3!}=79\\text{,}833\\text{,}600\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137543052\">There are 79,833,600 possible permutations of exam questions!<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165137805377\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137553609\">We can also use a calculator to find permutations. For this problem, we would enter 12, press the[latex]{\\,}_{n}{P}_{r}\\,[\/latex]\nfunction, enter 9, and then press the equal sign. The[latex]{\\,}_{n}{P}_{r}\\,[\/latex]\nfunction may be located under the MATH menu with probability commands.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137386959\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137939711\"><strong>Could we have solved <a class=\"autogenerated-content\" href=\"#Example_11_05_04\">(Figure)<\/a> using the Multiplication Principle?<\/strong><\/p>\n<p id=\"fs-id1165137442220\"><em>Yes. We could have multiplied<\/em>[latex]\\,15\\cdot 14\\cdot 13\\cdot 12\\cdot 11\\cdot 10\\cdot 9\\cdot 8\\cdot 7\\cdot 6\\cdot 5\\cdot 4\\,[\/latex]<em>to find the same answer<\/em>.<\/p>\n\n<\/div>\n<p id=\"eip-333\">A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.<\/p>\n\n<div id=\"eip-id1636733\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_06\">\n<div id=\"fs-id1165137444344\">\n<p id=\"fs-id1165137444345\">How many ways can the 7 actors line up?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135241065\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135241065\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135241065\"]\n<p id=\"fs-id1165135241066\">[latex]\\,P\\left(7,7\\right)=5,040\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1681009\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_07\">\n<div id=\"fs-id1165137603651\">\n<p id=\"fs-id1165137603652\">How many ways can 5 of the 7 actors be chosen to line up?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137933936\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137933936\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137933936\"]\n<p id=\"fs-id1165137933937\">[latex]\\,P\\left(7,5\\right)=2,520\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137648225\" class=\"bc-section section\">\n<h3>Find the Number of Combinations Using the Formula<\/h3>\n<p id=\"fs-id1165135193615\">So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of[latex]\\,r\\,[\/latex]objects from a set of[latex]\\,n\\,[\/latex]objects where the order does not matter can be written as[latex]\\,C\\left(n,r\\right).\\,[\/latex]Just as with permutations,[latex]\\,\\text{C}\\left(n,r\\right)\\,[\/latex]can also be written as[latex]{\\,}_{n}{C}_{r}.\\,[\/latex]In this case, the general formula is as follows.<\/p>\n\n<div id=\"eip-399\" class=\"unnumbered aligncenter\">[latex]\\,\\text{C}\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137654882\">An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are [latex]3!=3\u00b72\u00b71=6[\/latex] ways to order 3 paintings. There are [latex]\\frac{24}{6},\\,[\/latex]or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are <em>not<\/em> selecting 1 painting. There are 4 paintings we could choose <em>not<\/em> to select, so there are 4 ways to select 3 of the 4 paintings.<\/p>\n\n<div id=\"fs-id1165137731115\" class=\"textbox key-takeaways\">\n<h3>Formula for Combinations of <em>n<\/em> Distinct Objects<\/h3>\n<p id=\"fs-id1165137500953\">Given[latex]\\,n\\,[\/latex]distinct objects, the number of ways to select[latex]\\,r\\,[\/latex]objects from the set is<\/p>\n\n<div id=\"fs-id1165135531485\">[latex]\\,\\text{C}\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137433362\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137659772\"><strong>Given a number of options, determine the possible number of combinations.<\/strong><\/p>\n\n<ol id=\"fs-id1165137411159\" type=\"1\">\n \t<li>Identify[latex]\\,n\\,[\/latex]from the given information.<\/li>\n \t<li>Identify[latex]\\,r\\,[\/latex]from the given information.<\/li>\n \t<li>Replace[latex]\\,n\\,[\/latex]and[latex]\\,r\\,[\/latex]in the formula with the given values.<\/li>\n \t<li>Evaluate.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_05_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137666917\">\n<div id=\"fs-id1165137463613\">\n<h3>Finding the Number of Combinations Using the Formula<\/h3>\n<p id=\"fs-id1165137407735\">A fast food restaurant offers five side dish options. Your meal comes with two side dishes.<\/p>\n\n<ol id=\"fs-id1165137454885\" type=\"a\">\n \t<li>How many ways can you select your side dishes?<\/li>\n \t<li>How many ways can you select 3 side dishes?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137593658\" class=\"solution textbox shaded\">\n<div id=\"eip-id1165137667309\" class=\"unnumbered\">[reveal-answer q=\"632962\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"632962\"]\n<ol id=\"fs-id1165137593660\" type=\"a\">\n \t<li>We want to choose 2 side dishes from 5 options.\n<div id=\"eip-id1165137805004\" class=\"unnumbered\">[latex]\\,\\text{C}\\left(5,2\\right)=\\frac{5!}{2!\\left(5-2\\right)!}=10\\,[\/latex]<\/div><\/li>\n \t<li>We want to choose 3 side dishes from 5 options.\n<div id=\"eip-id1165137667309\" class=\"unnumbered\">[latex]\\,\\text{C}\\left(5,3\\right)=\\frac{5!}{3!\\left(5-3\\right)!}=10\\,[\/latex]<\/div><\/li>\n<\/ol>\n[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137465106\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137427098\">We can also use a graphing calculator to find combinations. Enter 5, then press[latex]{\\,}_{n}{C}_{r},\\,[\/latex]enter 3, and then press the equal sign. The[latex]{\\,}_{n}{C}_{r},\\,[\/latex]function may be located under the MATH menu with probability commands.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137558537\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137402033\"><strong>Is it a coincidence that parts (a) and (b) in <a class=\"autogenerated-content\" href=\"#Example_11_05_05\">(Figure)<\/a> have the same answers?<\/strong><\/p>\n<p id=\"fs-id1165137452885\"><em>No. When we choose r objects from n objects, we are <strong>not<\/strong> choosing[latex]\\,\\left(n\u2013r\\right)\\,[\/latex]objects. Therefore,[latex]\\,C\\left(n,r\\right)=C\\left(n,n\u2013r\\right).\\,[\/latex]<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165137457211\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_08\">\n<div id=\"fs-id1165137535204\">\n<p id=\"fs-id1165137535205\">An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137862737\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137862737\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137862737\"]\n<p id=\"fs-id1165137727648\">[latex]\\,C\\left(10,3\\right)=120\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438950\" class=\"bc-section section\">\n<h3>Finding the Number of Subsets of a Set<\/h3>\n<p id=\"fs-id1165135191446\">We have looked only at combination problems in which we chose exactly [latex]r[\/latex] objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?<\/p>\n<p id=\"fs-id1165137599617\">To answer this question, we need to consider pizzas with any number of toppings. There is [latex]C\\left(5,0\\right)=1[\/latex] way to order a pizza with no toppings. There are [latex]C\\left(5,1\\right)=5[\/latex] ways to order a pizza with exactly one topping. If we continue this process, we get<\/p>\n\n<div class=\"unnumbered\">[latex]\\,C\\left(5,0\\right)+C\\left(5,1\\right)+C\\left(5,2\\right)+C\\left(5,3\\right)+C\\left(5,4\\right)+C\\left(5,5\\right)=32\\,[\/latex]<\/div>\nThere are 32 possible pizzas. This result is equal to[latex]\\,{2}^{5}.\\,[\/latex]\n<p id=\"fs-id1165137473516\">We are presented with a sequence of choices. For each of the [latex]n[\/latex] objects we have two choices: include it in the subset or not. So for the whole subset we have made [latex]n\\,[\/latex] choices, each with two options. So there are a total of [latex]2\u00b72\u00b72\u00b7\\dots \u00b72[\/latex] possible resulting subsets, all the way from the empty subset, which we obtain when we say \u201cno\u201d each time, to the original set itself, which we obtain when we say \u201cyes\u201d each time.<\/p>\n\n<div id=\"fs-id1165135252247\" class=\"textbox key-takeaways\">\n<h3>Formula for the Number of Subsets of a Set<\/h3>\n<p id=\"fs-id1165137767403\">A set containing <em>n<\/em> distinct objects has [latex]{2}^{n}[\/latex] subsets.<\/p>\n\n<\/div>\n<div id=\"Example_11_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137767690\">\n<div>\n<h3>Finding the Number of Subsets of a Set<\/h3>\n<p id=\"fs-id1165137594426\">A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137532383\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137532383\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137532383\"]\n<p id=\"fs-id1165137532385\">We are looking for the number of subsets of a set with 4 objects. Substitute [latex]n=4[\/latex] into the formula.<\/p>\n\n<div id=\"eip-id1165134050650\" class=\"unnumbered\">[latex]\\,\\begin{array}{l}{2}^{n}={2}^{4}\\hfill \\\\ \\text{ }=16\\hfill \\end{array}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137901345\">There are 16 possible ways to order a potato.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605248\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_09\">\n<div id=\"fs-id1165137410962\">\n<p id=\"fs-id1165137410963\">A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137628240\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137628240\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137628240\"]\n<p id=\"fs-id1165137592089\">64 sundaes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137472540\" class=\"bc-section section\">\n<h3>Finding the Number of Permutations of <em>n<\/em> Non-Distinct Objects<\/h3>\n<p id=\"fs-id1165137532439\">We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be [latex]12![\/latex] ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the [latex]12![\/latex] permutations we counted are duplicates. The general formula for this situation is as follows.<\/p>\n\n<div id=\"eip-425\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137416703\">In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are [latex]4![\/latex] ways to order the stars and [latex]3![\/latex] ways to order the moon.<\/p>\n\n<div id=\"eip-660\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{12!}{4!3!}=3\\text{,}326\\text{,}400\\,[\/latex]<\/div>\n<p id=\"fs-id1165137442396\">There are 3,326,400 ways to order the sheet of stickers.<\/p>\n\n<div id=\"fs-id1165137442399\" class=\"textbox key-takeaways\">\n<h3>Formula for Finding the Number of Permutations of <em>n<\/em> Non-Distinct Objects<\/h3>\n<p id=\"fs-id1165137452296\">If there are [latex]n[\/latex] elements in a set and [latex]{r}_{1}\\,[\/latex]are alike,[latex]\\,{r}_{2}\\,[\/latex]are alike, [latex]{r}_{3}\\,[\/latex]are alike, and so on through [latex]{r}_{k},\\,[\/latex]the number of permutations can be found by<\/p>\n\n<div id=\"fs-id1165135203543\">[latex]\\,\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}\\,[\/latex]<\/div>\n<\/div>\n<div id=\"Example_11_05_07\" class=\"textbox examples\">\n<div id=\"fs-id1165135699142\">\n<div id=\"fs-id1165137436259\">\n<h3>Finding the Number of Permutations of <em>n<\/em> Non-Distinct Objects<\/h3>\n<p id=\"fs-id1165137532048\">Find the number of rearrangements of the letters in the word DISTINCT.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137466241\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137466241\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137466241\"]\n<p id=\"fs-id1165137450987\">There are 8 letters. Both I and T are repeated 2 times. Substitute[latex]\\,n=8, {r}_{1}=2, \\,[\/latex]and[latex]\\, {r}_{2}=2 \\,[\/latex]into the formula.<\/p>\n\n<div id=\"eip-id1165134371141\" class=\"unnumbered\">[latex]\\,\\frac{8!}{2!2!}=10\\text{,}080 \\,[\/latex]<\/div>\n<p id=\"fs-id1165137404564\">There are 10,080 arrangements.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160178\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_10\">\n<div id=\"fs-id1165137648088\">\n<p id=\"fs-id1165137648089\">Find the number of rearrangements of the letters in the word CARRIER.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135149841\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135149841\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135149841\"]\n<p id=\"fs-id1165135149842\">840<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462566\" class=\"precalculus media\">\n<p id=\"fs-id1165137507102\">Access these online resources for additional instruction and practice with combinations and permutations.<\/p>\n\n<ul>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/combinations\">Combinations<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/permutations\">Permutations<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898988\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165135178140\" summary=\"..\">\n<tbody>\n<tr>\n<td>number of permutations of[latex]\\,n\\,[\/latex]distinct objects taken[latex]\\,r\\,[\/latex]at a time<\/td>\n<td>[latex]P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>number of combinations of[latex]\\,n\\,[\/latex]distinct objects taken[latex]\\,r\\,[\/latex]at a time<\/td>\n<td>[latex]C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>number of permutations of[latex]\\,n\\,[\/latex]non-distinct objects<\/td>\n<td>[latex]\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137475706\">\n \t<li>If one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex]ways, then the first or second event can occur in [latex]m+n[\/latex] ways. See <a class=\"autogenerated-content\" href=\"#Example_11_05_01\">(Figure)<\/a>.<\/li>\n \t<li>If one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex]ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways. See <a class=\"autogenerated-content\" href=\"#Example_11_05_02\">(Figure)<\/a>.<\/li>\n \t<li>A permutation is an ordering of [latex]n[\/latex] objects.<\/li>\n \t<li>If we have a set of [latex]n[\/latex] objects and we want to choose [latex]r[\/latex] objects from the set in order, we write [latex]P\\left(n,r\\right).[\/latex]<\/li>\n \t<li>Permutation problems can be solved using the Multiplication Principle or the formula for [latex]P\\left(n,r\\right).[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_11_05_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_05_04\">(Figure)<\/a>.<\/li>\n \t<li>A selection of objects where the order does not matter is a combination.<\/li>\n \t<li>Given [latex]n[\/latex]distinct objects, the number of ways to select [latex]r[\/latex] objects from the set is [latex]\\text{C}\\left(n,r\\right)[\/latex] and can be found using a formula. See <a class=\"autogenerated-content\" href=\"#Example_11_05_05\">(Figure)<\/a>.<\/li>\n \t<li>A set containing [latex]n[\/latex] distinct objects has [latex]{2}^{n}[\/latex] subsets. See <a class=\"autogenerated-content\" href=\"#Example_11_05_06\">(Figure)<\/a>.<\/li>\n \t<li>For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See <a class=\"autogenerated-content\" href=\"#Example_11_05_07\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135638474\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135638477\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<p id=\"fs-id1165137634891\">For the following exercises, assume that there are [latex]n[\/latex] ways an event [latex]A[\/latex] can happen, [latex]m[\/latex] ways an event [latex]B[\/latex] can happen, and that [latex]A\\text{ and }B[\/latex] are non-overlapping.<\/p>\n\n<div id=\"fs-id1165137659838\">\n<div id=\"fs-id1165137659840\">\n<p id=\"fs-id1165137659841\">Use the Addition Principle of counting to explain how many ways event [latex]A\\text{ or }B[\/latex] can occur.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137543839\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137543839\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137543839\"]\n<p id=\"fs-id1165137543840\">There are[latex]\\,m+n\\,[\/latex]ways for either event[latex]\\,A\\,[\/latex]or event[latex]\\,B\\,[\/latex]to occur.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436857\">\n<div id=\"fs-id1165137436859\">\n<p id=\"fs-id1165137436860\">Use the Multiplication Principle of counting to explain how many ways event[latex]\\,A\\text{ and }B\\,[\/latex]can occur.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137735417\">Answer the following questions.<\/p>\n\n<div id=\"fs-id1165137735420\">\n<div id=\"fs-id1165137451089\">\n<p id=\"fs-id1165137451090\">When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137582620\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137582620\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137582620\"]\n<p id=\"fs-id1165137582622\">The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word \u201cor\u201d usually implies an addition problem. The word \u201cand\u201d usually implies a multiplication problem.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501159\">\n<div id=\"fs-id1165137416164\">\n<p id=\"fs-id1165137416166\">Describe how the permutation of[latex]n[\/latex] objects differs from the permutation of choosing [latex]r[\/latex] objects from a set of [latex]n[\/latex] objects. Include how each is calculated.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758968\">\n<div id=\"fs-id1165137758970\">\n<p id=\"fs-id1165137758972\">What is the term for the arrangement that selects [latex]r[\/latex] objects from a set of [latex]n[\/latex] objects when the order of the [latex]r[\/latex] objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137806120\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137806120\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137806120\"]\n<p id=\"fs-id1165137806121\">A combination;[latex]\\,C\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!r!}\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137646270\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137641893\">For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.<\/p>\n\n<div id=\"fs-id1165137643865\">\n<div id=\"fs-id1165137643867\">\n\nLet the set [latex]A=\\left\\{-5,-3,-1,2,3,4,5,6\\right\\}.[\/latex] How many ways are there to choose a negative or an even number from [latex]\\mathrm{A?}[\/latex]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137469306\">\n<div id=\"fs-id1165137469308\">\n<p id=\"fs-id1165137569701\">Let the set [latex]B=\\left\\{-23,-16,-7,-2,20,36,48,72\\right\\}.[\/latex] How many ways are there to choose a positive or an odd number from [latex]A?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135191547\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135191547\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135191547\"]\n<p id=\"fs-id1165135191548\">[latex]\\,4+2=6\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423222\">\n<div id=\"fs-id1165137423224\">\n<p id=\"fs-id1165137423225\">How many ways are there to pick a red ace or a club from a standard card playing deck?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137803944\">\n<div id=\"fs-id1165137803946\">\n<p id=\"fs-id1165137566060\">How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137566064\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137566064\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137566064\"]\n<p id=\"fs-id1165137407678\">[latex]\\,5+4+7=16\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137657858\">\n<div id=\"fs-id1165137657860\">\n<p id=\"fs-id1165135484160\">How many outcomes are possible from tossing a pair of coins?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135484165\">\n<div id=\"fs-id1165137851823\">\n<p id=\"fs-id1165137851824\">How many outcomes are possible from tossing a coin and rolling a 6-sided die?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135187115\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135187115\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135187115\"]\n<p id=\"fs-id1165135187116\">[latex]\\,2\u00d76=12\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874279\">\n<div id=\"fs-id1165137874281\">\n<p id=\"fs-id1165137453743\">How many two-letter strings\u2014the first letter from[latex]\\,A\\,[\/latex]and the second letter from[latex]\\,B\u2014[\/latex]can be formed from the sets[latex]\\,A=\\left\\{b,c,d\\right\\}\\,[\/latex]and[latex]\\,B=\\left\\{a,e,i,o,u\\right\\}?\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549744\">\n<div id=\"fs-id1165137549746\">\n<p id=\"fs-id1165137749653\">How many ways are there to construct a string of 3 digits if numbers can be repeated?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137749656\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137749656\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137749656\"]\n<p id=\"fs-id1165137749658\">[latex]\\,{10}^{3}=1000\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137455842\">\n<div id=\"fs-id1165135159869\">\n\nHow many ways are there to construct a string of 3 digits if numbers cannot be repeated?\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137925521\">For the following exercises, compute the value of the expression.<\/p>\n\n<div id=\"fs-id1165137925524\">\n<div>\n<p id=\"fs-id1165135190750\">[latex]\\,P\\left(5,2\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135191532\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135191532\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135191532\"]\n<p id=\"fs-id1165135191534\">[latex]\\,P\\left(5,2\\right)=20\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137593442\">\n<div id=\"fs-id1165135192212\">[latex]\\,P\\left(8,4\\right)\\,[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137759795\">\n<div id=\"fs-id1165137759797\">\n<p id=\"fs-id1165137759798\">[latex]\\,P\\left(3,3\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135194382\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135194382\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135194382\"]\n<p id=\"fs-id1165135194383\">[latex]\\,P\\left(3,3\\right)=6\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438758\">\n<div id=\"fs-id1165137438760\">\n<p id=\"fs-id1165137806524\">[latex]\\,P\\left(9,6\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137841491\">\n<div id=\"fs-id1165137841494\">\n<p id=\"fs-id1165137422912\">[latex]\\,P\\left(11,5\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137771782\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137771782\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137771782\"]\n<p id=\"fs-id1165137771784\">[latex]\\,P\\left(11,5\\right)=55,440\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736251\">\n<div id=\"fs-id1165137736253\">\n<p id=\"fs-id1165137762924\">[latex]\\,C\\left(8,5\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874584\">\n<div>\n<p id=\"fs-id1165137824360\">[latex]\\,C\\left(12,4\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137423187\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137423187\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137423187\"]\n<p id=\"fs-id1165137423188\">[latex]\\,C\\left(12,4\\right)=495\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137854863\">\n<div id=\"fs-id1165137529228\">\n<p id=\"fs-id1165137529229\">[latex]\\,C\\left(26,3\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137659864\">\n<div id=\"fs-id1165137838063\">\n<p id=\"fs-id1165137838064\">[latex]\\,C\\left(7,6\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137722860\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137722860\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137722860\"]\n<p id=\"fs-id1165137722861\">[latex]\\,C\\left(7,6\\right)=7\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135196857\">\n<div id=\"fs-id1165135196859\">\n<p id=\"fs-id1165135196860\">[latex]\\,C\\left(10,3\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137806510\">For the following exercises, find the number of subsets in each given set.<\/p>\n\n<div id=\"fs-id1165137806514\">\n<div>[latex]\\,\\left\\{1,2,3,4,5,6,7,8,9,10\\right\\}\\,[\/latex]<\/div>\n<div id=\"fs-id1165135570081\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135570081\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135570081\"]\n<p id=\"fs-id1165135570082\">[latex]\\,{2}^{10}=1024\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447686\">\n<div id=\"fs-id1165137447688\">\n<p id=\"fs-id1165137447689\">[latex]\\,\\left\\{a,b,c,\\dots ,z\\right\\}\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549605\">\n<div id=\"fs-id1165137451844\">\n<p id=\"fs-id1165137451845\">A set containing 5 distinct numbers, 4 distinct letters, and 3 distinct symbols<\/p>\n\n<\/div>\n<div id=\"fs-id1165137451849\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137451849\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137451849\"]\n<p id=\"fs-id1165137451850\">[latex]\\,{2}^{12}=4096\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137643752\">\n<p id=\"fs-id1165137643753\">The set of even numbers from 2 to 28<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137643757\">\n<div id=\"fs-id1165137871919\">\n<p id=\"fs-id1165137871921\">The set of two-digit numbers between 1 and 100 containing the digit 0<\/p>\n\n<\/div>\n<div id=\"fs-id1165137654642\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137654642\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137654642\"]\n<p id=\"fs-id1165137654644\">[latex]\\,{2}^{9}=512\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137628110\">For the following exercises, find the distinct number of arrangements.<\/p>\n\n<div id=\"fs-id1165135459875\">\n<div id=\"fs-id1165135459877\">\n<p id=\"fs-id1165135459878\">The letters in the word \u201cjuggernaut\u201d<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137848975\">\n<p id=\"fs-id1165137677635\">The letters in the word \u201cacademia\u201d<\/p>\n\n<\/div>\n<div id=\"fs-id1165137677638\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137677638\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137677638\"]\n<p id=\"fs-id1165137677639\">[latex]\\,\\frac{8!}{3!}=6720\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476778\">\n<div id=\"fs-id1165137755596\">\n<p id=\"fs-id1165137755597\">The letters in the word \u201cacademia\u201d that begin and end in \u201ca\u201d<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641802\">\n<div id=\"fs-id1165137641804\">\n<p id=\"fs-id1165137641805\">The symbols in the string #,#,#,@,@,$,$,$,%,%,%,%<\/p>\n\n<\/div>\n<div id=\"fs-id1165137641808\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137641808\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137641808\"]\n<p id=\"fs-id1165135185348\">[latex]\\,\\frac{12!}{3!2!3!4!}\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838339\">\n<div id=\"fs-id1165137838341\">\n<p id=\"fs-id1165137838343\">The symbols in the string #,#,#,@,@,$,$,$,%,%,%,% that begin and end with \u201c%\u201d<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832402\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137832407\">\n<div id=\"fs-id1165137548734\">\n\nThe set,[latex]\\,S\\,[\/latex]consists of[latex]\\,\\text{900,000,000}\\,[\/latex]whole numbers, each being the same number of digits long. How many digits long is a number from[latex]\\,S?\\,[\/latex](<em>Hint:<\/em> use the fact that a whole number cannot start with the digit 0.)\n\n<\/div>\n<div id=\"fs-id1165137936758\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137936758\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137936758\"]\n<p id=\"fs-id1165137936760\">9<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188310\">\n<div id=\"fs-id1165135188312\">\n<p id=\"fs-id1165135188313\">The number of 5-element subsets from a set containing[latex]\\,n\\,[\/latex]elements is equal to the number of 6-element subsets from the same set. What is the value of [latex]n?\\,[\/latex](<em>Hint:<\/em> the order in which the elements for the subsets are chosen is not important.)<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134056947\">\n<div id=\"fs-id1165134056949\">\n<p id=\"fs-id1165137731098\">Can [latex]C\\left(n,r\\right)[\/latex] ever equal [latex]P\\left(n,r\\right)?[\/latex] Explain.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137572461\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137572461\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137572461\"]\n<p id=\"fs-id1165137572462\">Yes, for the trivial cases [latex]r=0[\/latex] and [latex]r=1.[\/latex] If [latex]r=0,[\/latex] then [latex]C\\left(n,r\\right)=P\\left(n,r\\right)=1\\text{.\\hspace{0.17em}}[\/latex] If [latex]r=1,[\/latex] then [latex]r=1,[\/latex][latex]C\\left(n,r\\right)=P\\left(n,r\\right)=n.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137410377\">\n<p id=\"fs-id1165137704803\">Suppose a set [latex]A[\/latex] has 2,048 subsets. How many distinct objects are contained in [latex]A?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137772252\">\n<div id=\"fs-id1165137772254\">\n<p id=\"fs-id1165137772255\">How many arrangements can be made from the letters of the word \u201cmountains\u201d if all the vowels must form a string?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135528964\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135528964\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135528964\"]\n<p id=\"fs-id1165135528965\">[latex]\\,\\frac{6!}{2!}\u00d74!=8640\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135526953\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div>\n<div id=\"fs-id1165137694987\">\n\nA family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back.\n<ol type=\"a\">\n \t<li>How many arrangements are possible with no restrictions?<\/li>\n \t<li>How many arrangements are possible if the parents must sit in the front?<\/li>\n \t<li>How many arrangements are possible if the parents must be next to each other?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134373515\">\n<div id=\"fs-id1165134373517\">\n<p id=\"fs-id1165134056951\">A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134056956\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134056956\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134056956\"]\n<p id=\"fs-id1165134056957\">[latex]6-3+8-3=8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137409043\">\n<div id=\"fs-id1165137409046\">\n<p id=\"fs-id1165137409047\">In horse racing, a \u201ctrifecta\u201d occurs when a bettor wins by selecting the first three finishers in the exact order (1st place, 2nd place, and 3rd place). How many different trifectas are possible if there are 14 horses in a race?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660533\">\n<div id=\"fs-id1165137409049\">\n<p id=\"fs-id1165137409050\">A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137722285\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137722285\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137722285\"]\n<p id=\"fs-id1165137722286\">[latex]\\,4\u00d72\u00d75=40\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135572090\">\n<div id=\"fs-id1165135572092\">\n<p id=\"fs-id1165135572093\">Hector wants to place billboard advertisements throughout the county for his new business. How many ways can Hector choose 15 neighborhoods to advertise in if there are 30 neighborhoods in the county?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433487\">\n<div id=\"fs-id1165137433489\">\n<p id=\"fs-id1165137433490\">An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137724923\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137724923\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137724923\"]\n<p id=\"fs-id1165137724924\">[latex]\\,4\u00d712\u00d73=144\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134108402\">\n<div id=\"fs-id1165134108404\">\n<p id=\"fs-id1165134108405\">How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of[latex]\\,8\\,[\/latex]freshmen and[latex]\\,11\\,[\/latex]juniors?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137454186\">\n<div id=\"fs-id1165137454188\">\n<p id=\"fs-id1165135195564\">How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135195567\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135195567\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135195567\"]\n<p id=\"fs-id1165135195568\">[latex]\\,P\\left(15,9\\right)=1,816,214,400\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193138\">\n<div id=\"fs-id1165137433600\">\n<p id=\"fs-id1165137433601\">A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestra\u2019s 10 cellists and 16 violinists in order of musical proficiency. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735343\">\n<div id=\"fs-id1165137433604\">\n<p id=\"fs-id1165137433605\">A motorcycle shop has 10 choppers, 6 bobbers, and 5 caf\u00e9 racers\u2014different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 caf\u00e9 racers for a weekend showcase?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137673621\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137673621\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137673621\"]\n<p id=\"fs-id1165137673622\">[latex]C\\left(10,3\\right)\u00d7C\\left(6,5\\right)\u00d7C\\left(5,2\\right)=7,200[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n\nA skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137936941\">\n<div id=\"fs-id1165137936943\">\n<p id=\"fs-id1165137936944\">Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137803145\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137803145\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137803145\"]\n<p id=\"fs-id1165137803146\">[latex]\\,{2}^{11}=2048\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135207528\">\n<div>\n<p id=\"fs-id1165137454285\">A car wash offers the following optional services to the basic wash: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. How many washes are possible if any number of options can be added to the basic wash?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137454292\">\n<div id=\"fs-id1165137713907\">\n<p id=\"fs-id1165137713908\">Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137713912\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137713912\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137713912\"]\n<p id=\"fs-id1165137405697\">[latex]\\,\\frac{20!}{6!6!8!}=116,396,280\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137668309\">\n<div id=\"fs-id1165137668312\">\n<p id=\"fs-id1165137668313\">How many unique ways can a string of Christmas lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135255272\">\n \t<dt>Addition Principle<\/dt>\n \t<dd id=\"fs-id1165135255277\">if one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first or second event can occur in [latex]m+n[\/latex] ways<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137645178\">\n \t<dt>combination<\/dt>\n \t<dd id=\"fs-id1165135160429\">a selection of objects in which order does not matter<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135160433\">\n \t<dt>Fundamental Counting Principle<\/dt>\n \t<dd id=\"fs-id1165137668328\">if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways; also known as the Multiplication Principle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137651693\">\n \t<dt>Multiplication Principle<\/dt>\n \t<dd id=\"fs-id1165137651698\">if one event can occur in[latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways; also known as the Fundamental Counting Principle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137871530\">\n \t<dt>permutation<\/dt>\n \t<dd id=\"fs-id1165137551518\">a selection of objects in which order matters<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Solve counting problems using the Addition Principle.<\/li>\n<li>Solve counting problems using the Multiplication Principle.<\/li>\n<li>Solve counting problems using permutations involving n distinct objects.<\/li>\n<li>Solve counting problems using combinations.<\/li>\n<li>Find the number of subsets of a given set.<\/li>\n<li>Solve counting problems using permutations involving n non-distinct objects.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137432308\">A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!<\/p>\n<p id=\"fs-id1165135313702\">We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.<\/p>\n<div id=\"fs-id1165137463810\" class=\"bc-section section\">\n<h3>Using the Addition Principle<\/h3>\n<p id=\"fs-id1165137832066\">The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The <strong>Addition Principle<\/strong> tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options, as we can see in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_05_001\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_11_05_001\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154936\/CNX_Precalc_Figure_11_05_001n.jpg\" alt=\"The addition of 3 iPods and 4 iPhones.\" width=\"487\" height=\"358\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135186420\" class=\"textbox key-takeaways\">\n<h3>The Addition Principle<\/h3>\n<p id=\"fs-id1165135351541\">According to the Addition Principle, if one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first <em>or<\/em> second event can occur in [latex]m+n[\/latex] ways.<\/p>\n<\/div>\n<div id=\"Example_11_05_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137536266\">\n<div id=\"fs-id1165137552601\">\n<h3>Using the Addition Principle<\/h3>\n<p id=\"fs-id1165135206132\">There are 2 vegetarian entr\u00e9e options and 5 meat entr\u00e9e options on a dinner menu. What is the total number of entr\u00e9e options?<\/p>\n<\/div>\n<div id=\"fs-id1165137410849\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137399341\">We can add the number of vegetarian options to the number of meat options to find the total number of entr\u00e9e options.<\/p>\n<p><span id=\"fs-id1165135237185\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154938\/CNX_Precalc_Figure_11_05_002.jpg\" alt=\"The addition of the type of options for an entree.\" \/><\/span><\/p>\n<p id=\"fs-id1165137786497\">There are 7 total options.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843834\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_01\">\n<div id=\"fs-id1165137455516\">\n<p id=\"fs-id1165137529243\">A student is shopping for a new computer. He is deciding among 3 desktop computers and 4 laptop computers. What is the total number of computer options?<\/p>\n<\/div>\n<div id=\"fs-id1165137723942\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137732616\">7<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137645260\" class=\"bc-section section\">\n<h3>Using the Multiplication Principle<\/h3>\n<p id=\"fs-id1165137767730\">The <strong>Multiplication Principle<\/strong> applies when we are making more than one selection. Suppose we are choosing an appetizer, an entr\u00e9e, and a dessert. If there are 2 appetizer options, 3 entr\u00e9e options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in <a class=\"autogenerated-content\" href=\"#CNX_Precalculus_Figure_11_05_003\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalculus_Figure_11_05_003\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154940\/CNX_Precalc_Figure_11_05_003.jpg\" alt=\"A tree diagram of the different menu combinations.\" width=\"975\" height=\"287\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p>The possible choices are:<\/p>\n<ol id=\"fs-id1165137641640\" type=\"1\">\n<li>soup, chicken, cake<\/li>\n<li>soup, chicken, pudding<\/li>\n<li>soup, fish, cake<\/li>\n<li>soup, fish, pudding<\/li>\n<li>soup, steak, cake<\/li>\n<li>soup, steak, pudding<\/li>\n<li>salad, chicken, cake<\/li>\n<li>salad, chicken, pudding<\/li>\n<li>salad, fish, cake<\/li>\n<li>salad, fish, pudding<\/li>\n<li>salad, steak, cake<\/li>\n<li>salad, steak, pudding<\/li>\n<\/ol>\n<p>We can also find the total number of possible dinners by multiplying.<\/p>\n<p id=\"eip-933\">We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.<\/p>\n<div id=\"eip-828\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{llll}\\#\\text{ of appetizer options }\u00d7\\hfill & \\#\\text{ of entree options }\u00d7\\hfill & \\#\\text{ of dessert options}\\hfill & \\hfill \\\\ \\text{ }2\\text{ }\u00d7\\hfill & \\text{ }3\\text{ }\u00d7\\hfill & \\text{ }2\\hfill & =12\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165137560649\" class=\"textbox key-takeaways\">\n<h3>The Multiplication Principle<\/h3>\n<p id=\"fs-id1165137706934\">According to the Multiplication Principle, if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways. This is also known as the Fundamental Counting Principle.<\/p>\n<\/div>\n<div id=\"Example_11_05_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137771642\">\n<div id=\"fs-id1165134047731\">\n<h3>Using the Multiplication Principle<\/h3>\n<p id=\"fs-id1165137590306\">Diane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.<\/p>\n<\/div>\n<div id=\"fs-id1165137580422\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137771530\">To find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and the number of sweater options.<\/p>\n<p><span id=\"fs-id1165137733725\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154946\/CNX_Precalc_Figure_11_05_004.jpg\" alt=\"The multiplication of number of skirt options (2) times the number of blouse options (4) times the number of sweater options (2) which equals 16.\" \/><\/span><\/p>\n<p id=\"fs-id1165137755612\">There are 16 possible outfits.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137398102\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_02\">\n<div id=\"fs-id1165137571190\">\n<p id=\"fs-id1165137423732\">A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.<\/p>\n<\/div>\n<div id=\"fs-id1165137473501\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135255245\">There are 60 possible breakfast specials.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137553058\" class=\"bc-section section\">\n<h3>Finding the Number of Permutations of <em>n<\/em> Distinct Objects<\/h3>\n<p id=\"fs-id1165137461547\">The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.<\/p>\n<div id=\"fs-id1165137678383\" class=\"bc-section section\">\n<h4>Finding the Number of Permutations of <em>n<\/em> Distinct Objects Using the Multiplication Principle<\/h4>\n<p id=\"fs-id1165137725216\">To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.<\/p>\n<p><span id=\"fs-id1165137804112\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155013\/CNX_Precalc_Figure_11_05_005.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137732561\">There are four options for the first place, so we write a 4 on the first line.<\/p>\n<p><span id=\"fs-id1165137854788\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155021\/CNX_Precalc_Figure_11_05_006.jpg\" alt=\"Four times two blanks spots.\" \/><\/span><\/p>\n<p id=\"fs-id1165137597142\">After the first place has been filled, there are three options for the second place so we write a 3 on the second line.<\/p>\n<p><span id=\"fs-id1165137570729\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155023\/CNX_Precalc_Figure_11_05_007.jpg\" alt=\"Four times three times one blank spot.\" \/><\/span><\/p>\n<p id=\"fs-id1165137529516\">After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.<\/p>\n<p><span id=\"eip-id1165137874239\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155036\/CNX_Precalc_Figure_11_05_008.jpg\" alt=\"\" \/><\/span><\/p>\n<p>There are 24 possible permutations of the paintings.<\/p>\n<div id=\"fs-id1165137731358\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137436230\"><strong>Given<\/strong>[latex]\\,n\\,[\/latex]<strong>distinct options, determine how many permutations there are.<\/strong><\/p>\n<ol id=\"fs-id1165137552251\" type=\"1\">\n<li>Determine how many options there are for the first situation.<\/li>\n<li>Determine how many options are left for the second situation.<\/li>\n<li>Continue until all of the spots are filled.<\/li>\n<li>Multiply the numbers together.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_05_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137724829\">\n<div id=\"fs-id1165137551779\">\n<h3>Finding the Number of Permutations Using the Multiplication Principle<\/h3>\n<p>At a swimming competition, nine swimmers compete in a race.<\/p>\n<ol id=\"fs-id1165137803400\" type=\"a\">\n<li>How many ways can they place first, second, and third?<\/li>\n<li>How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)<\/li>\n<li>How many ways can all nine swimmers line up for a photo?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137533114\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165135177686\" type=\"a\">\n<li>Draw lines for each place.<br \/>\n<span id=\"eip-id1165133349418\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155038\/CNX_Precalc_Figure_11_05_009.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137827584\">There are 9 options for first place. Once someone has won first place, there are 8 remaining options for second place. Once first and second place have been won, there are 7 remaining options for third place.<\/p>\n<p><span id=\"eip-id1165135317102\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155040\/CNX_Precalc_Figure_11_05_010.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137533708\">Multiply to find that there are 504 ways for the swimmers to place.<\/p>\n<\/li>\n<li>Draw lines for describing each place.<br \/>\n<span id=\"eip-id1165133101712\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155042\/CNX_Precalc_Figure_11_05_011.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137527991\">We know Ariel must win first place, so there is only 1 option for first place. There are 8 remaining options for second place, and then 7 remaining options for third place.<\/p>\n<p><span id=\"eip-id1165137580736\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155049\/CNX_Precalc_Figure_11_05_012.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137464178\">Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first.<\/p>\n<\/li>\n<li>\n<p id=\"fs-id1165137531490\">Draw lines for describing each place in the photo.<\/p>\n<p><span id=\"eip-id1165135192171\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155051\/CNX_Precalc_Figure_11_05_013.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137779101\">There are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person remains for the last spot.<\/p>\n<p><span id=\"eip-id1165137551358\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155053\/CNX_Precalc_Figure_11_05_014.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137446031\">There are 362,880 possible permutations for the swimmers to line up.<\/p>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165137446031\"><\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165137896132\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135638483\">Note that in part c, we found there were 9! ways for 9 people to line up. The number of permutations of[latex]\\,n\\,[\/latex]distinct objects can always be found by[latex]\\,n!.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-327\">A family of five is having portraits taken. Use the Multiplication Principle to find the following.<\/p>\n<div id=\"eip-id2465210\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_03\">\n<div id=\"fs-id1165137817722\">\n<p id=\"fs-id1165137817723\">How many ways can the family line up for the portrait?<\/p>\n<\/div>\n<div id=\"fs-id1165137407672\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137407673\">120<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id2465241\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_04\">\n<div id=\"fs-id1165135193101\">\n<p id=\"fs-id1165135193102\">How many ways can the photographer line up 3 family members?<\/p>\n<\/div>\n<div id=\"fs-id1165137665821\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137665822\">60<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id2465272\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_05\">\n<div id=\"fs-id1165137810891\">\n<p id=\"fs-id1165137810892\">How many ways can the family line up for the portrait if the parents are required to stand on each end?<\/p>\n<\/div>\n<div id=\"fs-id1165137424182\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137424183\">12<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137698292\" class=\"bc-section section\">\n<h4>Finding the Number of Permutations of <em>n<\/em> Distinct Objects Using a Formula<\/h4>\n<p id=\"fs-id1165137862888\">For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let\u2019s look at two common notations for permutations. If we have a set of[latex]\\,n\\,[\/latex]objects and we want to choose[latex]\\,r\\,[\/latex]objects from the set in order, we write[latex]\\,P\\left(n,r\\right).\\,[\/latex]Another way to write this is [latex]{n}_{}{P}_{r},\\,[\/latex]a notation commonly seen on computers and calculators. To calculate[latex]\\,P\\left(n,r\\right),\\,[\/latex]we begin by finding[latex]\\,n!,\\,[\/latex]the number of ways to line up all [latex]n[\/latex] objects. We then divide by[latex]\\,\\left(n-r\\right)!\\,[\/latex] to cancel out the[latex]\\,\\left(n-r\\right)\\,[\/latex]items that we do not wish to line up.<\/p>\n<p id=\"fs-id1165137530985\">Let\u2019s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is [latex]6\u00d75\u00d74=120.[\/latex] Using factorials, we get the same result.<\/p>\n<div id=\"eip-959\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{6!}{3!}=\\frac{6\u00b75\u00b74\u00b73!}{3!}=6\u00b75\u00b74=120\\,[\/latex]<\/div>\n<p id=\"fs-id1165137406160\">There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.<\/p>\n<div id=\"eip-469\" class=\"unnumbered aligncenter\">[latex]\\,P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137443996\">Note that the formula stills works if we are choosing <u>all<\/u>[latex]\\,n\\,[\/latex]objects and placing them in order. In that case we would be dividing by[latex]\\,\\left(n-n\\right)!\\,[\/latex]or[latex]\\,0!,\\,[\/latex]which we said earlier is equal to 1. So the number of permutations of[latex]\\,n\\,[\/latex]objects taken[latex]\\,n\\,[\/latex]at a time is[latex]\\,\\frac{n!}{1}\\,[\/latex]or just[latex]\\,n!\\text{.}[\/latex]<\/p>\n<div id=\"fs-id1165137657370\" class=\"textbox key-takeaways\">\n<h3>Formula for Permutations of <em>n<\/em> Distinct Objects<\/h3>\n<p id=\"fs-id1165135503766\">Given[latex]\\,n\\,[\/latex]distinct objects, the number of ways to select[latex]\\,r\\,[\/latex]objects from the set in order is<\/p>\n<div>[latex]P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137644544\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137531306\"><strong>Given a word problem, evaluate the possible permutations.<\/strong><\/p>\n<ol id=\"fs-id1165137461265\" type=\"1\">\n<li>Identify[latex]\\,n\\,[\/latex]from the given information.<\/li>\n<li>Identify[latex]\\,r\\,[\/latex]from the given information.<\/li>\n<li>Replace[latex]\\,n\\,[\/latex]and[latex]\\,r\\,[\/latex]in the formula with the given values.<\/li>\n<li>Evaluate.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137660582\">\n<div id=\"fs-id1165137660584\">\n<h3>Finding the Number of Permutations Using the Formula<\/h3>\n<p id=\"fs-id1165137677568\">A professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and arrange the questions?<\/p>\n<\/div>\n<div id=\"fs-id1165137597587\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137597589\">Substitute[latex]\\,n=12\\,[\/latex]and[latex]\\,r=9\\,[\/latex]into the permutation formula and simplify.<\/p>\n<div id=\"eip-id1165134385701\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{ }P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}\\,\\hfill \\\\ P\\left(12,9\\right)=\\frac{12!}{\\left(12-9\\right)!}=\\frac{12!}{3!}=79\\text{,}833\\text{,}600\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137543052\">There are 79,833,600 possible permutations of exam questions!<\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id1165137805377\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137553609\">We can also use a calculator to find permutations. For this problem, we would enter 12, press the[latex]{\\,}_{n}{P}_{r}\\,[\/latex]<br \/>\nfunction, enter 9, and then press the equal sign. The[latex]{\\,}_{n}{P}_{r}\\,[\/latex]<br \/>\nfunction may be located under the MATH menu with probability commands.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137386959\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137939711\"><strong>Could we have solved <a class=\"autogenerated-content\" href=\"#Example_11_05_04\">(Figure)<\/a> using the Multiplication Principle?<\/strong><\/p>\n<p id=\"fs-id1165137442220\"><em>Yes. We could have multiplied<\/em>[latex]\\,15\\cdot 14\\cdot 13\\cdot 12\\cdot 11\\cdot 10\\cdot 9\\cdot 8\\cdot 7\\cdot 6\\cdot 5\\cdot 4\\,[\/latex]<em>to find the same answer<\/em>.<\/p>\n<\/div>\n<p id=\"eip-333\">A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.<\/p>\n<div id=\"eip-id1636733\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_06\">\n<div id=\"fs-id1165137444344\">\n<p id=\"fs-id1165137444345\">How many ways can the 7 actors line up?<\/p>\n<\/div>\n<div id=\"fs-id1165135241065\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135241066\">[latex]\\,P\\left(7,7\\right)=5,040\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1681009\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_07\">\n<div id=\"fs-id1165137603651\">\n<p id=\"fs-id1165137603652\">How many ways can 5 of the 7 actors be chosen to line up?<\/p>\n<\/div>\n<div id=\"fs-id1165137933936\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137933937\">[latex]\\,P\\left(7,5\\right)=2,520\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137648225\" class=\"bc-section section\">\n<h3>Find the Number of Combinations Using the Formula<\/h3>\n<p id=\"fs-id1165135193615\">So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of[latex]\\,r\\,[\/latex]objects from a set of[latex]\\,n\\,[\/latex]objects where the order does not matter can be written as[latex]\\,C\\left(n,r\\right).\\,[\/latex]Just as with permutations,[latex]\\,\\text{C}\\left(n,r\\right)\\,[\/latex]can also be written as[latex]{\\,}_{n}{C}_{r}.\\,[\/latex]In this case, the general formula is as follows.<\/p>\n<div id=\"eip-399\" class=\"unnumbered aligncenter\">[latex]\\,\\text{C}\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137654882\">An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are [latex]3!=3\u00b72\u00b71=6[\/latex] ways to order 3 paintings. There are [latex]\\frac{24}{6},\\,[\/latex]or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are <em>not<\/em> selecting 1 painting. There are 4 paintings we could choose <em>not<\/em> to select, so there are 4 ways to select 3 of the 4 paintings.<\/p>\n<div id=\"fs-id1165137731115\" class=\"textbox key-takeaways\">\n<h3>Formula for Combinations of <em>n<\/em> Distinct Objects<\/h3>\n<p id=\"fs-id1165137500953\">Given[latex]\\,n\\,[\/latex]distinct objects, the number of ways to select[latex]\\,r\\,[\/latex]objects from the set is<\/p>\n<div id=\"fs-id1165135531485\">[latex]\\,\\text{C}\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137433362\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137659772\"><strong>Given a number of options, determine the possible number of combinations.<\/strong><\/p>\n<ol id=\"fs-id1165137411159\" type=\"1\">\n<li>Identify[latex]\\,n\\,[\/latex]from the given information.<\/li>\n<li>Identify[latex]\\,r\\,[\/latex]from the given information.<\/li>\n<li>Replace[latex]\\,n\\,[\/latex]and[latex]\\,r\\,[\/latex]in the formula with the given values.<\/li>\n<li>Evaluate.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_05_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137666917\">\n<div id=\"fs-id1165137463613\">\n<h3>Finding the Number of Combinations Using the Formula<\/h3>\n<p id=\"fs-id1165137407735\">A fast food restaurant offers five side dish options. Your meal comes with two side dishes.<\/p>\n<ol id=\"fs-id1165137454885\" type=\"a\">\n<li>How many ways can you select your side dishes?<\/li>\n<li>How many ways can you select 3 side dishes?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137593658\" class=\"solution textbox shaded\">\n<div id=\"eip-id1165137667309\" class=\"unnumbered\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165137593660\" type=\"a\">\n<li>We want to choose 2 side dishes from 5 options.\n<div id=\"eip-id1165137805004\" class=\"unnumbered\">[latex]\\,\\text{C}\\left(5,2\\right)=\\frac{5!}{2!\\left(5-2\\right)!}=10\\,[\/latex]<\/div>\n<\/li>\n<li>We want to choose 3 side dishes from 5 options.\n<div id=\"eip-id1165137667309\" class=\"unnumbered\">[latex]\\,\\text{C}\\left(5,3\\right)=\\frac{5!}{3!\\left(5-3\\right)!}=10\\,[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137465106\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137427098\">We can also use a graphing calculator to find combinations. Enter 5, then press[latex]{\\,}_{n}{C}_{r},\\,[\/latex]enter 3, and then press the equal sign. The[latex]{\\,}_{n}{C}_{r},\\,[\/latex]function may be located under the MATH menu with probability commands.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137558537\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137402033\"><strong>Is it a coincidence that parts (a) and (b) in <a class=\"autogenerated-content\" href=\"#Example_11_05_05\">(Figure)<\/a> have the same answers?<\/strong><\/p>\n<p id=\"fs-id1165137452885\"><em>No. When we choose r objects from n objects, we are <strong>not<\/strong> choosing[latex]\\,\\left(n\u2013r\\right)\\,[\/latex]objects. Therefore,[latex]\\,C\\left(n,r\\right)=C\\left(n,n\u2013r\\right).\\,[\/latex]<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165137457211\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_08\">\n<div id=\"fs-id1165137535204\">\n<p id=\"fs-id1165137535205\">An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?<\/p>\n<\/div>\n<div id=\"fs-id1165137862737\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137727648\">[latex]\\,C\\left(10,3\\right)=120\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438950\" class=\"bc-section section\">\n<h3>Finding the Number of Subsets of a Set<\/h3>\n<p id=\"fs-id1165135191446\">We have looked only at combination problems in which we chose exactly [latex]r[\/latex] objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?<\/p>\n<p id=\"fs-id1165137599617\">To answer this question, we need to consider pizzas with any number of toppings. There is [latex]C\\left(5,0\\right)=1[\/latex] way to order a pizza with no toppings. There are [latex]C\\left(5,1\\right)=5[\/latex] ways to order a pizza with exactly one topping. If we continue this process, we get<\/p>\n<div class=\"unnumbered\">[latex]\\,C\\left(5,0\\right)+C\\left(5,1\\right)+C\\left(5,2\\right)+C\\left(5,3\\right)+C\\left(5,4\\right)+C\\left(5,5\\right)=32\\,[\/latex]<\/div>\n<p>There are 32 possible pizzas. This result is equal to[latex]\\,{2}^{5}.\\,[\/latex]<\/p>\n<p id=\"fs-id1165137473516\">We are presented with a sequence of choices. For each of the [latex]n[\/latex] objects we have two choices: include it in the subset or not. So for the whole subset we have made [latex]n\\,[\/latex] choices, each with two options. So there are a total of [latex]2\u00b72\u00b72\u00b7\\dots \u00b72[\/latex] possible resulting subsets, all the way from the empty subset, which we obtain when we say \u201cno\u201d each time, to the original set itself, which we obtain when we say \u201cyes\u201d each time.<\/p>\n<div id=\"fs-id1165135252247\" class=\"textbox key-takeaways\">\n<h3>Formula for the Number of Subsets of a Set<\/h3>\n<p id=\"fs-id1165137767403\">A set containing <em>n<\/em> distinct objects has [latex]{2}^{n}[\/latex] subsets.<\/p>\n<\/div>\n<div id=\"Example_11_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137767690\">\n<div>\n<h3>Finding the Number of Subsets of a Set<\/h3>\n<p id=\"fs-id1165137594426\">A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?<\/p>\n<\/div>\n<div id=\"fs-id1165137532383\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137532385\">We are looking for the number of subsets of a set with 4 objects. Substitute [latex]n=4[\/latex] into the formula.<\/p>\n<div id=\"eip-id1165134050650\" class=\"unnumbered\">[latex]\\,\\begin{array}{l}{2}^{n}={2}^{4}\\hfill \\\\ \\text{ }=16\\hfill \\end{array}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137901345\">There are 16 possible ways to order a potato.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605248\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_09\">\n<div id=\"fs-id1165137410962\">\n<p id=\"fs-id1165137410963\">A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?<\/p>\n<\/div>\n<div id=\"fs-id1165137628240\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137592089\">64 sundaes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137472540\" class=\"bc-section section\">\n<h3>Finding the Number of Permutations of <em>n<\/em> Non-Distinct Objects<\/h3>\n<p id=\"fs-id1165137532439\">We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be [latex]12![\/latex] ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the [latex]12![\/latex] permutations we counted are duplicates. The general formula for this situation is as follows.<\/p>\n<div id=\"eip-425\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165137416703\">In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are [latex]4![\/latex] ways to order the stars and [latex]3![\/latex] ways to order the moon.<\/p>\n<div id=\"eip-660\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{12!}{4!3!}=3\\text{,}326\\text{,}400\\,[\/latex]<\/div>\n<p id=\"fs-id1165137442396\">There are 3,326,400 ways to order the sheet of stickers.<\/p>\n<div id=\"fs-id1165137442399\" class=\"textbox key-takeaways\">\n<h3>Formula for Finding the Number of Permutations of <em>n<\/em> Non-Distinct Objects<\/h3>\n<p id=\"fs-id1165137452296\">If there are [latex]n[\/latex] elements in a set and [latex]{r}_{1}\\,[\/latex]are alike,[latex]\\,{r}_{2}\\,[\/latex]are alike, [latex]{r}_{3}\\,[\/latex]are alike, and so on through [latex]{r}_{k},\\,[\/latex]the number of permutations can be found by<\/p>\n<div id=\"fs-id1165135203543\">[latex]\\,\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}\\,[\/latex]<\/div>\n<\/div>\n<div id=\"Example_11_05_07\" class=\"textbox examples\">\n<div id=\"fs-id1165135699142\">\n<div id=\"fs-id1165137436259\">\n<h3>Finding the Number of Permutations of <em>n<\/em> Non-Distinct Objects<\/h3>\n<p id=\"fs-id1165137532048\">Find the number of rearrangements of the letters in the word DISTINCT.<\/p>\n<\/div>\n<div id=\"fs-id1165137466241\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137450987\">There are 8 letters. Both I and T are repeated 2 times. Substitute[latex]\\,n=8, {r}_{1}=2, \\,[\/latex]and[latex]\\, {r}_{2}=2 \\,[\/latex]into the formula.<\/p>\n<div id=\"eip-id1165134371141\" class=\"unnumbered\">[latex]\\,\\frac{8!}{2!2!}=10\\text{,}080 \\,[\/latex]<\/div>\n<p id=\"fs-id1165137404564\">There are 10,080 arrangements.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160178\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_05_10\">\n<div id=\"fs-id1165137648088\">\n<p id=\"fs-id1165137648089\">Find the number of rearrangements of the letters in the word CARRIER.<\/p>\n<\/div>\n<div id=\"fs-id1165135149841\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135149842\">840<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462566\" class=\"precalculus media\">\n<p id=\"fs-id1165137507102\">Access these online resources for additional instruction and practice with combinations and permutations.<\/p>\n<ul>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/combinations\">Combinations<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/permutations\">Permutations<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898988\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165135178140\" summary=\"..\">\n<tbody>\n<tr>\n<td>number of permutations of[latex]\\,n\\,[\/latex]distinct objects taken[latex]\\,r\\,[\/latex]at a time<\/td>\n<td>[latex]P\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>number of combinations of[latex]\\,n\\,[\/latex]distinct objects taken[latex]\\,r\\,[\/latex]at a time<\/td>\n<td>[latex]C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>number of permutations of[latex]\\,n\\,[\/latex]non-distinct objects<\/td>\n<td>[latex]\\frac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137475706\">\n<li>If one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex]ways, then the first or second event can occur in [latex]m+n[\/latex] ways. See <a class=\"autogenerated-content\" href=\"#Example_11_05_01\">(Figure)<\/a>.<\/li>\n<li>If one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex]ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways. See <a class=\"autogenerated-content\" href=\"#Example_11_05_02\">(Figure)<\/a>.<\/li>\n<li>A permutation is an ordering of [latex]n[\/latex] objects.<\/li>\n<li>If we have a set of [latex]n[\/latex] objects and we want to choose [latex]r[\/latex] objects from the set in order, we write [latex]P\\left(n,r\\right).[\/latex]<\/li>\n<li>Permutation problems can be solved using the Multiplication Principle or the formula for [latex]P\\left(n,r\\right).[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_11_05_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_05_04\">(Figure)<\/a>.<\/li>\n<li>A selection of objects where the order does not matter is a combination.<\/li>\n<li>Given [latex]n[\/latex]distinct objects, the number of ways to select [latex]r[\/latex] objects from the set is [latex]\\text{C}\\left(n,r\\right)[\/latex] and can be found using a formula. See <a class=\"autogenerated-content\" href=\"#Example_11_05_05\">(Figure)<\/a>.<\/li>\n<li>A set containing [latex]n[\/latex] distinct objects has [latex]{2}^{n}[\/latex] subsets. See <a class=\"autogenerated-content\" href=\"#Example_11_05_06\">(Figure)<\/a>.<\/li>\n<li>For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See <a class=\"autogenerated-content\" href=\"#Example_11_05_07\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135638474\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135638477\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<p id=\"fs-id1165137634891\">For the following exercises, assume that there are [latex]n[\/latex] ways an event [latex]A[\/latex] can happen, [latex]m[\/latex] ways an event [latex]B[\/latex] can happen, and that [latex]A\\text{ and }B[\/latex] are non-overlapping.<\/p>\n<div id=\"fs-id1165137659838\">\n<div id=\"fs-id1165137659840\">\n<p id=\"fs-id1165137659841\">Use the Addition Principle of counting to explain how many ways event [latex]A\\text{ or }B[\/latex] can occur.<\/p>\n<\/div>\n<div id=\"fs-id1165137543839\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137543840\">There are[latex]\\,m+n\\,[\/latex]ways for either event[latex]\\,A\\,[\/latex]or event[latex]\\,B\\,[\/latex]to occur.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436857\">\n<div id=\"fs-id1165137436859\">\n<p id=\"fs-id1165137436860\">Use the Multiplication Principle of counting to explain how many ways event[latex]\\,A\\text{ and }B\\,[\/latex]can occur.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137735417\">Answer the following questions.<\/p>\n<div id=\"fs-id1165137735420\">\n<div id=\"fs-id1165137451089\">\n<p id=\"fs-id1165137451090\">When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?<\/p>\n<\/div>\n<div id=\"fs-id1165137582620\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137582622\">The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word \u201cor\u201d usually implies an addition problem. The word \u201cand\u201d usually implies a multiplication problem.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501159\">\n<div id=\"fs-id1165137416164\">\n<p id=\"fs-id1165137416166\">Describe how the permutation of[latex]n[\/latex] objects differs from the permutation of choosing [latex]r[\/latex] objects from a set of [latex]n[\/latex] objects. Include how each is calculated.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758968\">\n<div id=\"fs-id1165137758970\">\n<p id=\"fs-id1165137758972\">What is the term for the arrangement that selects [latex]r[\/latex] objects from a set of [latex]n[\/latex] objects when the order of the [latex]r[\/latex] objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?<\/p>\n<\/div>\n<div id=\"fs-id1165137806120\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137806121\">A combination;[latex]\\,C\\left(n,r\\right)=\\frac{n!}{\\left(n-r\\right)!r!}\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137646270\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137641893\">For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.<\/p>\n<div id=\"fs-id1165137643865\">\n<div id=\"fs-id1165137643867\">\n<p>Let the set [latex]A=\\left\\{-5,-3,-1,2,3,4,5,6\\right\\}.[\/latex] How many ways are there to choose a negative or an even number from [latex]\\mathrm{A?}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137469306\">\n<div id=\"fs-id1165137469308\">\n<p id=\"fs-id1165137569701\">Let the set [latex]B=\\left\\{-23,-16,-7,-2,20,36,48,72\\right\\}.[\/latex] How many ways are there to choose a positive or an odd number from [latex]A?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135191547\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135191548\">[latex]\\,4+2=6\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423222\">\n<div id=\"fs-id1165137423224\">\n<p id=\"fs-id1165137423225\">How many ways are there to pick a red ace or a club from a standard card playing deck?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137803944\">\n<div id=\"fs-id1165137803946\">\n<p id=\"fs-id1165137566060\">How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?<\/p>\n<\/div>\n<div id=\"fs-id1165137566064\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137407678\">[latex]\\,5+4+7=16\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137657858\">\n<div id=\"fs-id1165137657860\">\n<p id=\"fs-id1165135484160\">How many outcomes are possible from tossing a pair of coins?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135484165\">\n<div id=\"fs-id1165137851823\">\n<p id=\"fs-id1165137851824\">How many outcomes are possible from tossing a coin and rolling a 6-sided die?<\/p>\n<\/div>\n<div id=\"fs-id1165135187115\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135187116\">[latex]\\,2\u00d76=12\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874279\">\n<div id=\"fs-id1165137874281\">\n<p id=\"fs-id1165137453743\">How many two-letter strings\u2014the first letter from[latex]\\,A\\,[\/latex]and the second letter from[latex]\\,B\u2014[\/latex]can be formed from the sets[latex]\\,A=\\left\\{b,c,d\\right\\}\\,[\/latex]and[latex]\\,B=\\left\\{a,e,i,o,u\\right\\}?\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549744\">\n<div id=\"fs-id1165137549746\">\n<p id=\"fs-id1165137749653\">How many ways are there to construct a string of 3 digits if numbers can be repeated?<\/p>\n<\/div>\n<div id=\"fs-id1165137749656\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137749658\">[latex]\\,{10}^{3}=1000\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137455842\">\n<div id=\"fs-id1165135159869\">\n<p>How many ways are there to construct a string of 3 digits if numbers cannot be repeated?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137925521\">For the following exercises, compute the value of the expression.<\/p>\n<div id=\"fs-id1165137925524\">\n<div>\n<p id=\"fs-id1165135190750\">[latex]\\,P\\left(5,2\\right)\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135191532\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135191534\">[latex]\\,P\\left(5,2\\right)=20\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137593442\">\n<div id=\"fs-id1165135192212\">[latex]\\,P\\left(8,4\\right)\\,[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137759795\">\n<div id=\"fs-id1165137759797\">\n<p id=\"fs-id1165137759798\">[latex]\\,P\\left(3,3\\right)\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135194382\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135194383\">[latex]\\,P\\left(3,3\\right)=6\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438758\">\n<div id=\"fs-id1165137438760\">\n<p id=\"fs-id1165137806524\">[latex]\\,P\\left(9,6\\right)\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137841491\">\n<div id=\"fs-id1165137841494\">\n<p id=\"fs-id1165137422912\">[latex]\\,P\\left(11,5\\right)\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137771782\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137771784\">[latex]\\,P\\left(11,5\\right)=55,440\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736251\">\n<div id=\"fs-id1165137736253\">\n<p id=\"fs-id1165137762924\">[latex]\\,C\\left(8,5\\right)\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874584\">\n<div>\n<p id=\"fs-id1165137824360\">[latex]\\,C\\left(12,4\\right)\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137423187\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137423188\">[latex]\\,C\\left(12,4\\right)=495\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137854863\">\n<div id=\"fs-id1165137529228\">\n<p id=\"fs-id1165137529229\">[latex]\\,C\\left(26,3\\right)\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137659864\">\n<div id=\"fs-id1165137838063\">\n<p id=\"fs-id1165137838064\">[latex]\\,C\\left(7,6\\right)\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137722860\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137722861\">[latex]\\,C\\left(7,6\\right)=7\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135196857\">\n<div id=\"fs-id1165135196859\">\n<p id=\"fs-id1165135196860\">[latex]\\,C\\left(10,3\\right)\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137806510\">For the following exercises, find the number of subsets in each given set.<\/p>\n<div id=\"fs-id1165137806514\">\n<div>[latex]\\,\\left\\{1,2,3,4,5,6,7,8,9,10\\right\\}\\,[\/latex]<\/div>\n<div id=\"fs-id1165135570081\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135570082\">[latex]\\,{2}^{10}=1024\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447686\">\n<div id=\"fs-id1165137447688\">\n<p id=\"fs-id1165137447689\">[latex]\\,\\left\\{a,b,c,\\dots ,z\\right\\}\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549605\">\n<div id=\"fs-id1165137451844\">\n<p id=\"fs-id1165137451845\">A set containing 5 distinct numbers, 4 distinct letters, and 3 distinct symbols<\/p>\n<\/div>\n<div id=\"fs-id1165137451849\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137451850\">[latex]\\,{2}^{12}=4096\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137643752\">\n<p id=\"fs-id1165137643753\">The set of even numbers from 2 to 28<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137643757\">\n<div id=\"fs-id1165137871919\">\n<p id=\"fs-id1165137871921\">The set of two-digit numbers between 1 and 100 containing the digit 0<\/p>\n<\/div>\n<div id=\"fs-id1165137654642\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137654644\">[latex]\\,{2}^{9}=512\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137628110\">For the following exercises, find the distinct number of arrangements.<\/p>\n<div id=\"fs-id1165135459875\">\n<div id=\"fs-id1165135459877\">\n<p id=\"fs-id1165135459878\">The letters in the word \u201cjuggernaut\u201d<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137848975\">\n<p id=\"fs-id1165137677635\">The letters in the word \u201cacademia\u201d<\/p>\n<\/div>\n<div id=\"fs-id1165137677638\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137677639\">[latex]\\,\\frac{8!}{3!}=6720\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476778\">\n<div id=\"fs-id1165137755596\">\n<p id=\"fs-id1165137755597\">The letters in the word \u201cacademia\u201d that begin and end in \u201ca\u201d<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641802\">\n<div id=\"fs-id1165137641804\">\n<p id=\"fs-id1165137641805\">The symbols in the string #,#,#,@,@,$,$,$,%,%,%,%<\/p>\n<\/div>\n<div id=\"fs-id1165137641808\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135185348\">[latex]\\,\\frac{12!}{3!2!3!4!}\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838339\">\n<div id=\"fs-id1165137838341\">\n<p id=\"fs-id1165137838343\">The symbols in the string #,#,#,@,@,$,$,$,%,%,%,% that begin and end with \u201c%\u201d<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832402\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137832407\">\n<div id=\"fs-id1165137548734\">\n<p>The set,[latex]\\,S\\,[\/latex]consists of[latex]\\,\\text{900,000,000}\\,[\/latex]whole numbers, each being the same number of digits long. How many digits long is a number from[latex]\\,S?\\,[\/latex](<em>Hint:<\/em> use the fact that a whole number cannot start with the digit 0.)<\/p>\n<\/div>\n<div id=\"fs-id1165137936758\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137936760\">9<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188310\">\n<div id=\"fs-id1165135188312\">\n<p id=\"fs-id1165135188313\">The number of 5-element subsets from a set containing[latex]\\,n\\,[\/latex]elements is equal to the number of 6-element subsets from the same set. What is the value of [latex]n?\\,[\/latex](<em>Hint:<\/em> the order in which the elements for the subsets are chosen is not important.)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134056947\">\n<div id=\"fs-id1165134056949\">\n<p id=\"fs-id1165137731098\">Can [latex]C\\left(n,r\\right)[\/latex] ever equal [latex]P\\left(n,r\\right)?[\/latex] Explain.<\/p>\n<\/div>\n<div id=\"fs-id1165137572461\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137572462\">Yes, for the trivial cases [latex]r=0[\/latex] and [latex]r=1.[\/latex] If [latex]r=0,[\/latex] then [latex]C\\left(n,r\\right)=P\\left(n,r\\right)=1\\text{.\\hspace{0.17em}}[\/latex] If [latex]r=1,[\/latex] then [latex]r=1,[\/latex][latex]C\\left(n,r\\right)=P\\left(n,r\\right)=n.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137410377\">\n<p id=\"fs-id1165137704803\">Suppose a set [latex]A[\/latex] has 2,048 subsets. How many distinct objects are contained in [latex]A?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137772252\">\n<div id=\"fs-id1165137772254\">\n<p id=\"fs-id1165137772255\">How many arrangements can be made from the letters of the word \u201cmountains\u201d if all the vowels must form a string?<\/p>\n<\/div>\n<div id=\"fs-id1165135528964\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135528965\">[latex]\\,\\frac{6!}{2!}\u00d74!=8640\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135526953\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div>\n<div id=\"fs-id1165137694987\">\n<p>A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back.<\/p>\n<ol type=\"a\">\n<li>How many arrangements are possible with no restrictions?<\/li>\n<li>How many arrangements are possible if the parents must sit in the front?<\/li>\n<li>How many arrangements are possible if the parents must be next to each other?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134373515\">\n<div id=\"fs-id1165134373517\">\n<p id=\"fs-id1165134056951\">A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?<\/p>\n<\/div>\n<div id=\"fs-id1165134056956\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134056957\">[latex]6-3+8-3=8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137409043\">\n<div id=\"fs-id1165137409046\">\n<p id=\"fs-id1165137409047\">In horse racing, a \u201ctrifecta\u201d occurs when a bettor wins by selecting the first three finishers in the exact order (1st place, 2nd place, and 3rd place). How many different trifectas are possible if there are 14 horses in a race?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660533\">\n<div id=\"fs-id1165137409049\">\n<p id=\"fs-id1165137409050\">A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?<\/p>\n<\/div>\n<div id=\"fs-id1165137722285\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137722286\">[latex]\\,4\u00d72\u00d75=40\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135572090\">\n<div id=\"fs-id1165135572092\">\n<p id=\"fs-id1165135572093\">Hector wants to place billboard advertisements throughout the county for his new business. How many ways can Hector choose 15 neighborhoods to advertise in if there are 30 neighborhoods in the county?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433487\">\n<div id=\"fs-id1165137433489\">\n<p id=\"fs-id1165137433490\">An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?<\/p>\n<\/div>\n<div id=\"fs-id1165137724923\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137724924\">[latex]\\,4\u00d712\u00d73=144\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134108402\">\n<div id=\"fs-id1165134108404\">\n<p id=\"fs-id1165134108405\">How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of[latex]\\,8\\,[\/latex]freshmen and[latex]\\,11\\,[\/latex]juniors?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137454186\">\n<div id=\"fs-id1165137454188\">\n<p id=\"fs-id1165135195564\">How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?<\/p>\n<\/div>\n<div id=\"fs-id1165135195567\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135195568\">[latex]\\,P\\left(15,9\\right)=1,816,214,400\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193138\">\n<div id=\"fs-id1165137433600\">\n<p id=\"fs-id1165137433601\">A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestra\u2019s 10 cellists and 16 violinists in order of musical proficiency. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735343\">\n<div id=\"fs-id1165137433604\">\n<p id=\"fs-id1165137433605\">A motorcycle shop has 10 choppers, 6 bobbers, and 5 caf\u00e9 racers\u2014different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 caf\u00e9 racers for a weekend showcase?<\/p>\n<\/div>\n<div id=\"fs-id1165137673621\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137673622\">[latex]C\\left(10,3\\right)\u00d7C\\left(6,5\\right)\u00d7C\\left(5,2\\right)=7,200[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>A skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137936941\">\n<div id=\"fs-id1165137936943\">\n<p id=\"fs-id1165137936944\">Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?<\/p>\n<\/div>\n<div id=\"fs-id1165137803145\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137803146\">[latex]\\,{2}^{11}=2048\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135207528\">\n<div>\n<p id=\"fs-id1165137454285\">A car wash offers the following optional services to the basic wash: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. How many washes are possible if any number of options can be added to the basic wash?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137454292\">\n<div id=\"fs-id1165137713907\">\n<p id=\"fs-id1165137713908\">Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?<\/p>\n<\/div>\n<div id=\"fs-id1165137713912\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137405697\">[latex]\\,\\frac{20!}{6!6!8!}=116,396,280\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137668309\">\n<div id=\"fs-id1165137668312\">\n<p id=\"fs-id1165137668313\">How many unique ways can a string of Christmas lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135255272\">\n<dt>Addition Principle<\/dt>\n<dd id=\"fs-id1165135255277\">if one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first or second event can occur in [latex]m+n[\/latex] ways<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137645178\">\n<dt>combination<\/dt>\n<dd id=\"fs-id1165135160429\">a selection of objects in which order does not matter<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135160433\">\n<dt>Fundamental Counting Principle<\/dt>\n<dd id=\"fs-id1165137668328\">if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways; also known as the Multiplication Principle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137651693\">\n<dt>Multiplication Principle<\/dt>\n<dd id=\"fs-id1165137651698\">if one event can occur in[latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\u00d7n[\/latex] ways; also known as the Fundamental Counting Principle<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137871530\">\n<dt>permutation<\/dt>\n<dd id=\"fs-id1165137551518\">a selection of objects in which order matters<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":6,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-209","chapter","type-chapter","status-publish","hentry"],"part":198,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/209","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/209\/revisions"}],"predecessor-version":[{"id":210,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/209\/revisions\/210"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/198"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/209\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=209"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=209"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=209"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}