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# a2 - S x as a product of polynomials so that x k]Φ S x...

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MATH239 Assignment 2 Due: May 20 Your assignment must be in the appropriate drop box by noon Friday. 1. (4 points) Let S be a set of conﬁgurations with weight function w . Show that, for each non-negative integer n , [ x n ] Φ S ( x ) 1 - x is equal to the number of elements of S of weight n . 2. (9 points) Consider choosing marbles from a bag containing 3 red marbles, 6 white marbles, and 4 green marbles; marbles of the same colour are indistinguishable. (a) Determine a generating function Φ S ( x ), as a product of polynomials, so that [ x k S ( x ) counts the number of ways of choosing k marbles. (b) Determine a generating function Φ S ( x ) so that [ x k S ( x ) counts the number of ways of choosing k marbles under the condition that we must choose at least one marble of each colour. (c) Determine a generating function Φ
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Unformatted text preview: S ( x ), as a product of polynomials, so that [ x k ]Φ S ( x ) counts the number of ways of choosing k marbles under the condition that the total number of red and green marbles chosen is odd. 3. (4 points) Suppose that the formal power series A ( x ) = a + a 1 x + a 2 x 2 + ··· satisﬁes the equation (1 + x + x 2 ) A ( x ) = 3 + 5 x 2 . By considering the coeﬃcients of x , x 1 , x 2 , and x 3 on either side of this equation, determine a , a 1 , a 2 , and a 3 . 4. (8 points) (a) Using Theorem 1.8.2, show that (1-4 x ) 1 2 = 1-2 x X k ≥ 1 k + 1 ± 2 k k ! x k . (b) Using the Binomial Theorem (Theorem 1.8.1), show that (1-4 x )-1 2 = X k ≥ ± 2 k k ! x k ....
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