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Unformatted text preview: MATH 239 Assignment 1 Solutions This assignment is due at noon on Friday, May 15, 2009, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. Let integers 0 ≤ ` ≤ k ≤ n be given. Consider the binomial identity n k k ` = n ` n ` k ` . (a) Give an algebraic proof of this identity. Solution: Using the formula ( n k ) = n ! ( n k )! k ! we find LHS = n ! ( n k )! k ! k ! ( k ` )! ` ! = n ! ( n k )!( k ` )! ` ! and RHS = n ! ( n ` )! ` ! ( n ` )! ( n k )!( k ` )! = n ! ( n k )!( k ` )! ` ! , so LHS=RHS as required. (b) Give a combinatorial proof of this identity. Solution: We define a set S of pairs of subsets as follows: S = { ( A, B ) : A ⊆ B ⊆ { 1 , 2 , . . . n } ,  A  = `,  B  = k } . We calculate the size of the set S in two different ways. First, we choose k elements of { 1 , 2 , . . . , n } in all possible ways to form the set B , then from the k elements of B we choose ` elements in all possible ways to form the set A . Therefore the number of elements of S is ( n k )( k ` ) . Now we calculate the size of S by choosing the set A first. We can do this by choosing ` elements of { 1 , 2 , . . . , n } in all possible ways. Then to form the set B , we must choose k ` elements from the remaining n ` elements of { 1 , 2 , . . . , n } that were not chosen to be elements of A . There are therefore ( n ` k ` ) such choices. Therefore the size of S is ( n ` )( n ` k ` ) . Thus we have ( n k )( k ` ) =  S  = ( n ` )( n ` k ` ) as required....
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This note was uploaded on 08/09/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
 Spring '09
 M.PEI
 Combinatorics

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