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475finalsolfall07

# 475finalsolfall07 - MATH 475 Final Exam 140...

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MATH 475; Final Exam, 140 points, December 19, 2007 (R.A.Brualdi) Name: These R. Solutions Do not multiply out factorials, combination numbers, etc. Circle your an- swers. I. (10+5=15 points) Let h n be the number of n -permutation of A’s, B’s C’s, D’s, and E’s where there are an even number of A’s, an odd number of B’s, at least one C, with no restriction on the number of D’s and E’s. (1) Find a simple closed formula for the exponential generating function of h 0 , h 1 , h 2 , . . . . e x + e - x 2 e x - e - x 2 ( e x - 1) e x e x = · · · = e 5 x - e 4 x - e x + 1 4 (2) Find a simple closed formula for h n . h n = 5 n - 4 n - 1 2 if n 1 0 if n = 0 II. (30 points, 3 points each) Give something counted by each of the following numbers. Possible answers are: 1. k n : The number of n -permutations of k objects with unlimited repetition. 2. n !: The number of permutations of n distinct objects. 3. n k : The number of subsets (committees) of size k of an n element set. 4. k n k : The number of committees of size k of an n element set with one of the elements distinguished (e.g. leader). 5. n k =0 n k 2 : The number of n -combinations of a set of 2 n elements. 6. n ! n k =0 ( - 1) k k !

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475finalsolfall07 - MATH 475 Final Exam 140...

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