722 CHAPTER 11 Probability for the third position. Thus, there are nine possibilities for the fourth position. Since the fifth position cannot be filled by any of the two digits previously used, there are eight possibilities for the fifth position. The last position can be filled by any of the seven remaining digits. 5 L 25 L 9 D 9 D 8 D 7 D Since 5 25 9 9 8 7 567,000, ⋅ ⋅ ⋅ ⋅ ⋅ = there are 567,000 different arrangements that meet the conditions specified. 7 Now try Exercise 29 Example 2 Fundamental Counting Principle: Classic Rock Albums At the Paramount Swap Meet, Jerry wishes to display his five most popular classic rock albums in line on his table. The artists of these albums are Aerosmith, Boston, Kiss, Queen, and Styx. a) In how many different ways can he display the five albums? b) If he wants to place the Styx album in the middle, in how many different ways can he arrange the albums? c) If he wants the Kiss album to be first and the Queen album to be last, in how many different ways can he arrange the albums? Solution a) There are five positions to fill, using the five albums. In the first position, on the left, he can use any one of the five albums. In the second position, he can use any of the four remaining albums. In the third position, he can use any of the three remaining albums, and so on. The number of distinct possible arrangements is 5 4 3 2 1 120 ⋅ ⋅ ⋅ ⋅ = b) We begin by satisfying the specified requirements stated. In this case, the Styx album must be placed in the middle. Therefore, there is only one possibility for the middle position. 1 For the first position, there are now four possibilities. For the second position, there will be three possibilities. For the fourth position, there will be two possibilities. Finally, in the last position, there is only one possibility. 4 3 1 2 1 24 ⋅ ⋅ ⋅ ⋅ = Thus, under the condition stated, there are 24 different possible arrangements. c) For the first album, there is only one possibility, the Kiss album. For the last album, there is only one possibility, the Queen album. 1 1 The second position can be filled by any of the three remaining albums. The third position can be filled by any of the two remaining albums. There is only one album left for the fourth position. Thus, the number of possible arrangements is 1 3 2 1 1 6 ⋅ ⋅ ⋅ ⋅ = There are only six possible arrangements that satisfy the given conditions. 7 Now try Exercise 31 Blueee77/Shutterstock
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