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Dr. A. Betten Fall 2005 M501 Combinatorics I exercise sheet # 2 Exercise # 1 Prove the following identities: n k n n−l a) = , k l l k−l (3 points) k X m n m+n b) = (recall that i k−i k i=0 c) k X n+i i i=0 n X = n k = 0 for k < 0 or k > n), n+k+1 , k 2 n 0 (−1) d) = m (−1) k k=0 k if n is odd if n = 2m. 2m m Exercise # 2 Let Fn be the n-th Fibonacci number. Prove the following identities: (3 points) a) Fn2 − Fn+1 Fn−1 = (−1)n for n ≥ 1. b) n X Fi = Fn+2 − 1. i=0 2 + Fn2 = F2n , c) Fn−1 bn/2c d) Fn = X i=0 Fn−1 Fn + Fn Fn+1 = F2n+1 . n−i . i Exercise # 3 (1 points) Let Fn be the n-th Fibonacci number. Show that Fn is composite for all odd n > 3. Exercise # 4 (3 points) If n is a multiple of 8, show that the number of subsets of an n-set of size divisible by 4 is 2n−2 + 2(n−2)/2 . Example: for n = 8, we have 80 + 84 + 88 = 1 + 70 + 1 = 72 = 64 + 8 = 26 + 23 . √ Hint: consider (1 + i)n , where i = −1. due to Friday, 9/23/05.