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

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Tuesday, June 28 Math 239 Midterm Questions 1. Use the Binomial Theorem to prove that: (a) n X k = 0 ˆ n k ! = 2 n (b) n X k = 0 k ˆ n k ! = n 2 n - 1 Solution: (a) According to the Binomial Theorem, n X k = 0 ˆ n k ! x k = (1 + x ) n . (1) Let x = 1, we get n k = 0 ( n k ) = 2 n . (b) Differentiate both sides of Equation 1 to get n X k = 0 ˆ n k ! kx k - 1 = n (1 + x ) n - 1 . (2) Let x = 1, we get n k = 0 ( 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 0 }. Let S = S k 0 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 0 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).
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Math 239 Midterm Questions 3. The generating function for the set Ω of binary strings that do contain a copy of 0110 is 1 + x 3 1 - 2 x + x 3 - x 4 .
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