quiz-3-solutions

quiz-3-solutions - MATH 239 Quiz Solutions Spring 2010...

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MATH 239 Quiz Solutions — Spring 2010 — Section 3 1. Let k and n be nonnegative integers. Determine the following coefficient as a summation: [ x n ](1 - 2 x ) 20 (1 - x 2 ) - k . Solution. [ x n ](1 - 2 x ) 20 (1 - x 2 ) - k = [ x n ] X i 0 ( - 2) i ± 20 i ² x i X j 0 ± j + k - 1 k - 1 ² x 2 j = [ x n ] X i 0 X j 0 ( - 2) i ± 20 i ²± j + k - 1 k - 1 ² x i +2 j = b n/ 2 c X j =0 ( - 2) n - 2 j ± 20 n - 2 j ²± j + k - 1 k - 1 ² , where we let i = n - 2 j . Alternatively, we may index the final summation as X { j 0: n - 2 j 0 } , X { j 0:20 n - 2 j 0 } or b n/ 2 c X j =max { 0 , d ( n - 20) / 2 e} . 2. At an intergalactic yard sale, there are three distinct planets costing 5, 7 and 9 gold coins respectively, one comet costing 12 gold coins, and 120 identical stars selling for 1 gold coin each. For a positive integer n , how many ways can one spend n gold coins in this sale? Write your answer as the coefficient in a rational function. Solution.
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This note was uploaded on 11/26/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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quiz-3-solutions - MATH 239 Quiz Solutions Spring 2010...

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