assgt1 - MATH 239 ASSIGNMENT 1 Due Friday September 19 at...

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Unformatted text preview: MATH 239 ASSIGNMENT 1 Due Friday, September 19, at NOON in drop boxes (at St. Jerome’s for Sec 01, outside MC 4067 for Sec 02, 03, 04) 1 (a) Give a combinatorial proof of the identity p m + n k P = k s i =0 p m i Pp n k-i P , where k, m, n are nonnegative integers. (b) Give an algebraic proof of the identity in part (a). 2. Let a n,k be the number of k-element subsets of { 1 , . . . , n } that have no consecutive pairs of elements ( i.e. , i and i + 1 form a consecutive pair for any i = 1 , . . . , n-1). Give a combinatorial proof that a n,k = p n-k + 1 k P , where k, n are nonnegative integers with n ≥ 2 k-1. 3. Let S ( n, m ) = ∑ n i =0 (-1) i ( n i )( n m-i ) , for nonnegative integers m, n . Give an algebraic proof that S ( n, m ) = 0 if m is odd. Evaluate S ( n, m ) when m is even. 4. Let S be the set of subsets of { 1 , 2 , 3 , 4 } . We de±ne three weight functions for S : for s ∈ S , let ω 1 ( s ) be the sum of the elements in s (where an empty sum is 0); let ω 2 ( s ) be...
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