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Unformatted text preview: Tuesday, June 28 Math 239 Midterm Questions 1. Use the Binomial Theorem to prove that: (a) n X k = ˆ n k ! = 2 n (b) n X k = k ˆ n k ! = n 2 n 1 Solution: (a) According to the Binomial Theorem, n X k = ˆ n k ! x k = (1 + x ) n . (1) Let x = 1, we get ∑ n k = ( n k ) = 2 n . (b) Differentiate both sides of Equation 1 to get n X k = ˆ n k ! kx k 1 = n (1 + x ) n 1 . (2) Let x = 1, we get ∑ n k = ( n k ) k = n 2 n 1 . 2. Find the generating function for the number of compositions of n with an even number of parts, each of which is congruent to 1 modulo 3. Solution: Let A = {3 a + 1 : a ∈ N ≥ }. Let S = S k ≥ A 2 k . Let the weight function on A be α ( r ) = r and let the weight function on S be the sum of the weight of the components. The coefficient of x n in Φ S ( x ) is then the number of compositions of n with an even number of parts, each of which is congruent to 1 modulo 3. We have Φ A ( x ) = x + x 4 +···= x 1 x 3 . By the sum and product lemma, Φ S ( x ) = X k ≥ Φ A ( x ) 2 k = 1 1 Φ A ( x ) 2 = (1 x 3 ) 2 (1 x 3 ) 2 x 2 = 1 2 x 3 + x 6 1 x 2 2 x 3 + x 6 . The k = 0 term is included since by definition, there is a single composition with 0 parts (which is even). Math 239 Midterm Questions 3. The generating function for the set Ω...
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This note was uploaded on 08/13/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
 Spring '09
 M.PEI
 Binomial Theorem, Binomial

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