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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 nonattacking rooks on an nbyn
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 ndimensional 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.
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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.
 Fall '09
 WILDSTROM
 Integers

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