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# hw4 - HOMEWORK#4 DUE ON 10/18(1(Chapter 6 Exercise 33)1 Let...

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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 ofwaysto place k non-attackingrookson an n -by- n board,sothateachofthemisinoneofthepositions(1 , 1),(2 , 2),...,( n, n ) and (1 , 2), (2 , 3), ..., ( n 1 , n ), ( n, 1). Prove that a ( n, k )= 2 n 2 n k parenleftbigg 2 n k k parenrightbigg . (2) Find the number of bijections σ : [ n ] [ n ], such that σ (1) negationslash∈ { 1 , 2 } , σ (2) negationslash∈ { 1 , 2 } , σ (3) negationslash =2, and σ (5) negationslash∈ { 4 , 5 } . (3) Let Q n =([ n ] , ) be the n -dimensional hypercube. Let f : P ([ n ]) R be a function such that for all K [ n
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