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Unformatted text preview: Due: Friday noon, May 27. Each question carries 20 marks. Math 239 Assignment 3 1. How many nonnegative integers smaller than 100,000 have the sum of their digits equal to 23? 2. Consider compositions of a positive integer n into 2 positive parts. (a) Define an appropriate set S 1 and write down a generating function S 1 ( x ) to count the number of compositions n = c + d with c < d . Express your answer as a rational function. (b) Define another appropriate set S 2 and write down another generating function S 2 ( x ) to count the number of compositions n = e + f with even e and arbitrary f . Express your answer as a rational function. (c) Suppose S 1 ( x ) = n 1 a n x n and S 2 ( x ) = n 1 b n x n . Use your answers in (a) and (b) to show that n a n = b n . Express a n in terms of n . (d) For each n , let S 1 n and S 2 n be the respective subsets of S 1 and S 2 containing the compo- sitions of n . Establish a bijection between S 1 n and S 2 n (which maps any (...
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This note was uploaded on 05/26/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
- Spring '09