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08 - Fall 2009 CS131 Combinatorial Structures Homework 8...

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Fall 2009 CS131 – Combinatorial Structures Homework 8 Homework 8, due Nov 17 You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check ex- ercises of the easier kind in the book, try to solve them, too! Problem 1 (LPV 3.8.4) . (10pts) Find all values of n and k for which ( n k + 1 ) = 3 ( n k ) . Solution. We have n k + 1 = n ( n - 1) ··· ( n - k + 1)( n - k ) k !( k + 1) , n k = n ( n - 1) ··· ( n - k + 1) k ! . The equality is therefore n ( n - 1) ··· ( n - k + 1)( n - k ) k !( k + 1) = 3 n ( n - 1) ··· ( n - k + 1) k ! , n - k k + 1 = 3 . Rearrangement gives n = 4 k + 3. The equation holds for k = 0 , 1 , 2 ,... , with n = 4 k + 3. Problem 2. (10pts) Find a closed formula for the following sum: z 3 - z 5 + z 7 - z 9 +··· , where | z | < 1. Solution. This is an infinite geometric series that can be written as z 3 (1 + q + q 2 +··· ) , where q = - z 2 . Therefore the result is z 3 1 - q = z 3 1 + z 2 . Problem 3 (see LPV 3.6.3) . (10pts) Show the following identity by combinatorial reasoning: n 0 2 n n + n 1 2 n n + 1 +···+ n n 2 n 2 n = 3 n 2 n .
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Fall 2009 CS131 – Combinatorial Structures Homework 8 Solution.
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