midsolns - Tuesday, June 28 Math 239 Midterm Questions 1....

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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 = n k ! = 2 n (b) n X k = k n k ! = n 2 n- 1 Solution: (a) According to the Binomial Theorem, n X k = 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 = 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 ( r ) = r and let the weight function on S be the sum of the weight of the components. The coefficient of x n in 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 A ( x ) = x + x 4 += x 1- x 3 . By the sum and product lemma, S ( x ) = X k A ( x ) 2 k = 1 1- 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...
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midsolns - Tuesday, June 28 Math 239 Midterm Questions 1....

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