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# HW5 - HW5 Solutions Dilip Raghavan 1 Section 6.1 Problem...

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HW5 Solutions Dilip Raghavan October 14, 2008 1 Section 6.1 Problem 4. (a) We want all solution to the equation e 1 + e 2 + e 3 + e 4 + e 5 = r with 0 e i 5 So the generating function is (1 + x + x 2 + x 3 + x 4 + x 5 ) 5 (b) We want all solutions to the equation e 1 + e 2 + e 3 + e 4 = r with 3 e i 6. So the generating function is ( x 3 + x 4 + x 5 + x 6 ) 4 (c) We want all solutions to the equation e 1 + e 2 + e 3 + e 4 + e 5 + e 6 + e 7 = r with e i 1. So the generating function is k = 1 x k 7 = x 1 - x 7 (d) We want solutions to e 1 + e 2 + e 3 = r with e 1 5. So the generating function is (1 + x + x 2 + x 3 + x 4 + x 5 ) × k = 0 x k 2 = (1 + x + x 2 + x 3 + x 4 + x 5 ) × 1 1 - x 2 Problem 14. (a) We want solutions to the equation e 1 + e 2 + e 3 + e 4 + e 5 + e 6 = r with e 1 , e 2 and e 3 odd and the e 4 , e 5 and e 6 even. So the generating function is ( x + x 3 + x 5 ) 3 ( x 2 + x 4 + x 6 ) 3 (b) We want solutions to e 1 + e 2 + e 3 + e 4 + e 5 + e 6 = r with e i i . So the generating function is

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