hw4 - HOMEWORK #4, DUE ON 10/18 (1) (Chapter 6, Exercise...

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Unformatted text preview: HOMEWORK #4, DUE ON 10/18 (1) (Chapter 6, Exercise 33)1 Let n and k be positive integers with k ≤ n. Let a(n, k ) be the number of ways to place k non-attacking rooks on an n-by-n board, so that each of them is in one of the positions (1, 1), (2, 2), . . . , (n, n) and (1, 2), (2, 3), . . . , (n − 1, n), (n, 1). Prove that a(n, k ) = 2n 2n − k . 2n − k k (2) Find the number of bijections σ : [n] → [n], such that σ (1) ∈ {1, 2}, σ (2) ∈ {1, 2}, σ (3) = 2, and σ (5) ∈ {4, 5}. (3) Let Qn = ([n], ⊆) be the n-dimensional hypercube. Let f : P ([n]) → R be a function such that for all K ⊆ [n], f (L) = L ⊆K 5 2 |K | . Find a simple formula for f ([n]) (with no summation). The next two problems are from my “marital disharmony” series. (4) How many ways can n married couple stand in a row in such a way that nobody stands next to their spouse. (5) How many ways can n married couple shake hands simultaneously, if nobody shakes hands with their spouse. 1 Of course I can’t resist assigning the *’d question. 1 ...
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

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