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Unformatted text preview: 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 As, Bs Cs, Ds, and Es where there are an even number of As, an odd number of Bs, at least one C, with no restriction on the number of Ds and Es. (1) Find a simple closed formula for the exponential generating function of h , 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 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....
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