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qsola.09

# qsola.09 - MATH 239 Quiz No calculators or other aids may...

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MATH 239 Quiz No calculators or other aids may be used. Show all your work. 1. Let S = V × Y × Z where V is the set of all positive odd integers Y is the set { 4 , 8 , 12 , 16 , . . . , 200 } Z is the set of all non-negative even integers. Find the generating function for S with respect to the weight function w defined by w ( v, y, z ) = v + y + ( z/ 2). Express your answer in the form p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials in x . Solution: the generating function for V with respect to the weight function w ( v ) = v is x + x 3 + x 5 + . . . = x (1 - x 2 ) - 1 . the generating function for Y with respect to the weight function w ( y ) = y is x 4 + x 8 + x 12 + . . . + x 200 = x 4 (1 + x 4 + x 8 + . . . + x 196 ) = x 4 1 - x 200 1 - x 4 . the generating function for Z with respect to the weight function w ( z ) = z/ 2 is 1 + x + x 2 + x 3 + . . . = (1 - x ) - 1 . Since S is a Cartesian product, and the conditions for the Product Lemma are satisfied, we find the generating function for S is Φ S ( x ) = x 1 - x 2 x 4 (1 - x 200 ) 1 - x 4 1 1 - x = x 5 (1 - x 200 ) (1 - x 2 )(1 - x 4 )(1 - x ) = x 5 - x 205 (1 - x 2 - x 4 + x 6 )(1 - x ) = x 5 - x 205 (1 - x 2 - x 4 + x 6 ) - x (1 - x 2 - x 4 + x 6 ) = x 5 - x 205

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qsola.09 - MATH 239 Quiz No calculators or other aids may...

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