726 CHAPTER 11 Probability In Example 5, we determined that when selecting three of five letters, there were 60 permutations. We can obtain the same result using the permutation formula: P 5! (5 3)! 5! 2! 5 4 3 2 1 2 1 60 5 3 = − = = ⋅ ⋅ ⋅ ⋅ ⋅ = Permutation Formula The number of permutations possible when r objects are selected from n objects is determined by the permutation formula P n n r ! ( )! n r = − Example 6 Fishing Club Officers The fishing club at Central New Mexico Community College has nine members. To determine club officers, nine slips of paper each containing the name of a member are placed into a hat. The club advisor randomly selects a first name from the hat to become the president, a second name to become secretary, and a third name to become treasurer. How many different arrangements of club officers are possible? Solution There are nine people, n 9, = of which three are to be selected; thus, r 3. = P 9! (9 3)! 9! 6! 9 8 7 6 5 4 3 2 1 6 5 4 3 2 1 504 9 3 = − = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = Thus, with nine people there can be 504 different arrangements for president, secretary, and treasurer. 7 Now try Exercise 43 In Example 6, the fraction 9 8 7 6 5 4 3 2 1 6 5 4 3 2 1 504 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = can be also expressed as 9 8 7 6! 6! 504 ⋅ ⋅ ⋅ = The solution to Example 6, like other permutation problems, can also be obtained using the fundamental counting principle. Example 7 Bicycle Club The Rainbow Bicycle Club has 10 different full-day routes that members wish to travel exactly once, but they have only 6 specific dates for their trips. In how many ways can the different routes be assigned to the dates scheduled for their trips? Solution There are 10 possible routes but only 6 specific dates scheduled for the trips. Since traveling route A on day 1 and traveling route B on day 2 is different than traveling route B on day 1 and traveling route A on day 2, we have a permutation problem. There are 10 possible routes; thus, n 10. = There are 6 routes that are going to be selected and assigned to different days; thus, r 6. = Now we
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