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Discrete Mathematics with Graph Theory (3rd Edition) 203

Discrete Mathematics with Graph Theory (3rd Edition) 203 -...

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Section 7.6 7' (1 - .1. + .1. - .1. + .1. _ .1. + .1. _ .1.) . I! 2! 3! 4! 5! 6! 7! 7! - 7! + 7 . 6 . 5 . 4 . 3 - 7 . 6 . 5 . 4 + 7 . 6 . 5 - 7 . 6 + 7 - 1 2520 - 840 + 210 - 42 + 6 = 1854 Ds 8' (1 - .1. + .1. _ .1. + .1. _ .1. + .1. _ .1. + .1.) . I! 2! 3! 4! 5! 6! 7! S! 8! - 8! + 8 . 7 . 6 . 5 . 4 . 3 - 8 . 7 . 6 . 5 . 4 + 8 . 7 . 6 . 5 - 8 . 7 . 6 +8· 7 - 8 + 1 = 20,160 - 6720 + 1680 - 336 + 56 - 7 = 14,833 2. This is D 26 = 26!(1- fr + tr - -lr + ... + 2~!)' 3. [BB] This is Dn = 11!(1 - fr + -ft - -lr + ... - 1~!)' 4. D50 5. (a) D7 6. (a) D 20 (b) [BB] 7! - D7 (b) 20! - D 20 (c) 1 201 (c) [BB] There are 20 choices for the person who receives his or her own hat. For each choice, the 19 other people can get their hats in D 19 ways. Hence, the answer is 20D 19 . (d) This is the answer to (b) less the answer to (c); that is, 20! - D 20 - 20D 19 . (e) This means either no person receives hislher own hat or exactly one person receives hislher own hat or exactly two people receive their own hats. The answer is D 20 + 20D 19 + e2 0 )D 1S . 7. (a) [BB] Let Al be the set of permutations of 1-9 such that 1 is in position 1; A3 the set of perm uta- tions of 1-9 such that 3 is in position 3. Define A5, A7 and A9 similarly. Then I Ui Ail = Li IAil- Li<i IAi nAil + Li<i<k IAi n Ai n Akl - Li<i<k<t IAi n Ai n Ak n Atl + IAI n A3 n A5 n A7 n A91 = 5(8!) - (~)7! + (~)6! - (~)5! + 4!.
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