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# Remember that the vowels in v can be arranged two

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Remember that the vowels in V can be arranged two ways. To give an

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8 example, suppose V = I U and V 1 = O. The four ways this can occur are as follows: I U O; U I O; O I U; O U I. The number of arrangements with all three vowels together and our two specified vowels together is 8! 2 2 ! ! x 4 = 8!. The number of arrangements with exactly two vowels together is C(3,2)x[ 9! 2! - 8!] . The number of arrangements with at least two vowels together is this answer plus the one we obtained in part a. Divide this by 10! 2 2 ! ! to get the fraction of such arrangements. Problems from page 176 6. There are n ways to pick the man. Once the man is picked there are n-1 women who are not his wife. Pick one of them in n-1 ways. By the fundamental principle of counting, there are n(n-1) ways to pick a man and woman who are not husband and wife. 7. a. There are 20 books in total so there are 20 ways in which one book could be selected. b. 8x7x5. From the eight English books pick one of them, from the seven French books pick one, and from the five German books pick one of them. c. There are 6 cases to consider here. i. Two French and one English. ii. Two French and one German. iii. Two German and one English. iv. Two German and one French. v. Two English and one German. vi. Two English and one French. i. Select two French books in C(7,2) ways and one English book in 8 ways. Once they are selected, there are 3! ways to arrange them. C(7,2)x8x3! = 1008.
9 ii. Select two French books in C(7,2) ways and one German book in 5 ways. Once they are selected, there are 3! ways to arrange them. C(7,2)x5x3! = 630. iii. Select two German books in C(5,2) ways and one English book in 8 ways. Once they are selected, there are 3! ways to arrange them. C(5,2)x8x3! = 480. iv. Select two German books in C(5,2) ways and one French book in 7 ways. Once they are selected, there are 3! ways to arrange them. C(5,2)x7x3! = 420. v. Select two English books in C(8,2) ways and one German book in 5 ways. Once they are selected, there are 3! ways to arrange them. C(8,2)x5x3! = 840. vi. Select two English books in C(8,2) ways and one French book in 7 ways. Once they are selected, there are 3! ways to arrange them. C(8,2)x7x3! = 1176. Adding these all up, we get 4554. 10. 3 A’s and 5B’s. A collection of A’s and B’s can only be distinguished by the number of A’s and B’s in the collection. Any collection can have 0,1,2, or 3 A’s, or 0,1,2,3,4, or 5 B’s. There are 4 x 6 -1 =23 non empty subcollections of A’s and B’s. 13. a. The first digit can’t be zero, so we only have 9 choices for this digit. We have 10 choices for each of the other digits. Therefore, there are 9x10 4 five digit numbers. b. A number is even if and only if the last digit is even. There are 5 choices for the last digit, and 9 choices for the first digit. After taking care of these restrictions, the other three digits can be selected in 10 ways. There are 9x5x10 3 = 45000 even five digit numbers.

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• Fall '06
• miller
• Numerical digit, ways

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