184a_99w_fe - Math 184A Final Exam 811AM Monday 1 AUTOMATIC...

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Math 184A Final Exam 8–11AM Monday 12/13/99 1. AUTOMATIC contains 2 pairs of repeated letters and 7 distinct letters for a total of 9 letters. For 4 n 7, find a formula for the number of n -letter “words” that can be made from this collection of letters. Your answer should be a formula involving n , not 4 separate numbers for n = 4 , 5 , 6 , 7. 2. Let P k ( n ) be the number of permutations of the set { 1 , . . . , n } having no cycles of length greater than k . Thus P 1 ( n ) = 1 for n > 0 since all cycles are of length 1. For later convenience, define P k ( n ) = 0 , if n < 0 1 , if n = 0. (a) By considering the cycle containing n +1, prove that P 2 ( n +1) = P 2 ( n )+ nP 2 ( n - 1) for n 0. (Be careful for small values of n .) (b) State and prove a similar recursion for P 3 ( n + 1). 3. Define a “special” tree to be a rooted plane tree which is either a single vertex (the root) or a root vertex that is joined to either a left tree or a right tree or both a left and right tree.
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