Exam (elaborations) TEST BANK FOR A First Course In Probability 7th Edition By Sheldon Ross (Solution Manual)

1. (a) By the generalized basic principle of counting there are 26 ⋅ 26 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 = 67,600,000 (b) 26 ⋅ 25 ⋅ 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 = 19,656,000 2. 64 = 1296 3. An assignment is a sequence i1, …, i20 where ij is the job to which person j is assigned. Since only one person can be assigned to a job, it follows that the sequence is a permutation of the numbers 1, …, 20 and so there are 20! different possible assignments. 4. There are 4! possible arrangements. By assigning instruments to Jay, Jack, John and Jim, in that order, we see by the generalized basic principle that there are 2 ⋅ 1 ⋅ 2 ⋅ 1 = 4 possibilities. 5. There were 8 ⋅ 2 ⋅ 9 = 144 possible codes. There were 1 ⋅ 2 ⋅ 9 = 18 that started with a 4. 6. Each kitten can be identified by a code number i, j, k, l where each of i, j, k, l is any of the numbers from 1 to 7. The number i represents which wife is carrying the kitten, j then represents which of that wife’s 7 sacks contain the kitten; k represents which of the 7 cats in sack j of wife i is the mother of the kitten; and l represents the number of the kitten of cat k in sack j of wife i. By the generalized principle there are thus 7 ⋅ 7 ⋅ 7 ⋅ 7 = 2401 kittens 7. (a) 6! = 720 (b) 2 ⋅ 3! ⋅ 3! = 72 (c) 4!3! = 144 (d) 6 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 1 ⋅ 1 = 72 8. (a) 5! = 120 (b) 2!2! 7! = 1260 (c) 4!4!2! 11! = 34,650 (d) 2!2! 7! = 1260 9. 6!4! (12)! = 27,720 10. (a) 8! = 40,320 (b) 2 ⋅ 7! = 10,080 (c) 5!4! = 2,880 (d) 4!24 = 384 2 Chapter 1 11. (a) 6! (b) 3!2!3! (c) 3!4! 12. (a) 305 (b) 30 ⋅ 29 ⋅ 28 ⋅ 27 ⋅ 26 13.      2 20 14.      5 52 15. There are          5 12 5 10 possible choices of the 5 men and 5 women. They can then be paired up in 5! ways, since if we arbitrarily order the men then the first man can be paired with any of the 5 women, the next with any of the remaining 4, and so on. Hence, there are        5 12 5 5! 10 possible results. 16. (a)     +      +       2 4 2 7 2 6 = 42 possibilities. (b) There are 6 ⋅ 7 choices of a math and a science book, 6 ⋅ 4 choices of a math and an economics book, and 7 ⋅ 4 choices of a science and an economics book. Hence, there are 94 possible choices. 17. The first gift can go to any of the 10 children, the second to any of the remaining 9 children, and so on. Hence, there are 10 ⋅ 9 ⋅ 8 ⋅ ⋅ ⋅ 5 ⋅ 4 = 604,800 possibilities. 18.              3 4 2 6 2 5 = 600 19. (a) There are             +           2 4 1 2 3 8 3 4 3 8 = 896 possible committees. There are          3 4 3 8 that do not contain either of the 2 men, and there are              2 4 1 2 3 8 that contain exactly 1 of them. (b) There are             +          3 6 2 6 1 2 3 6 3 6 = 1000 possible committees.