11.8 Combinations 735 We have presented various counting methods, including the fundamental counting principle, permutations, and combinations. You often need to decide which method to use to solve a problem. Table 11.4 may help you in selecting the procedure to use. Solution a) Since the order in which the team members are selected does not matter, this is a combination problem. Isaac must choose four players from 6 eligible students. C 6! (6 4)!4! 6! 2!4! 6 5 4 3 2 1 2 1 4 3 2 1 30 2 15 6 4 = − = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = Thus, there are 15 different teams of four players possible. b) Since the order in which the pizza toppings are selected does not matter, this is a combination problem. There are 11 toppings, and they must choose 3. C 11! (11 3)!3! 11! 8!3! 1110987654321 87654321321 990 6 165 11 3 = − = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ = = Thus, there are 165 different 3-topping pizzas possible. 7 Now try Exercise 29 Learning Catalytics Keyword: Angel-SOM-11.8 (See Preface for additional details.) Example 4 Dinner Combinations At the Tokyo Bangkok restaurant, dinner consists of selecting 3 items from column A, selecting 4 items from column B, and selecting 3 items from column C from the menu. If columns A, B, and C have 5, 7, and 6 items, respectively, how many different dinner combinations are possible? Solution For column A, 3 of 5 items must be selected, which can be represented as C . 5 3 For column B, 4 of 7 items must be selected, which can be represented as C . 7 4 For column C, 3 of 6 items must be selected, or C . 6 3 C C C 10 35 and 20 5 3 7 4 6 3 = = = Using the fundamental counting principle, we can determine the total number of dinner combinations by multiplying the number of choices from columns A, B, and C: C C C Total number of dinner choices 10 35 20 7000 5 3 7 4 6 3 = ⋅ ⋅ = ⋅ ⋅ = Therefore, 7000 different combinations are possible under these conditions. 7 Now try Exercise 41 Table 11.4 Summary of Counting Methods Fundamental Counting Principle: If a first experiment has M distinct outcomes and a second experiment has N distinct outcomes, then the two experiments in that specific order have M N⋅ distinct outcomes. The fundamental counting principle may be used with or without repetition of items. It is used when determining the number of distinct outcomes when two or more experiments are performed. It is also used when there are specific placement requirements, such as the first digit must be a 0 or 1. Determining the Number of Ways of Selecting r Items from n Items (Repetition Is Not Permitted) Permutations Permutations are used when order is important. For example, a, b, c and b, c, a are two different permutations of the same three letters. P n n r ! ( )! n r = − Problems solved with the permutation formula may also be solved by using the fundamental counting principle. Combinations Combinations are used when order is not important. For example, a, b, c and b, c, a are the same combination of three letters. But a, b, c, and a, b, d are two different combinations of three letters. C n n r r ! ( )! ! n r = − Instructor Resources for Section 11.8 in MyLab Math • Objective-Level Videos 11.8 • Interactive Concept Video: Counting Methods • Interactive Concept Video: Counting Playing Cards • Interactive Concept Video: Counting Problems with “Not”, “Or”, “At Least One” • PowerPoint Lecture Slides 11.8 • MyLab Exercises and Assignments 11.8
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