quiz3-1 - EECS 310 Fall 2009 Quiz#3 Instructor Nicole...

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EECS 310 - Fall 2009: Quiz #3 Instructor: Nicole Immorlica Date: Nov 24, 2009 Answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. You must answer all questions. Name: 1. (20 points) Combinatorics. (a) (10 points) Give a combinatorial proof of the fact that ( n k ) = n k ( n - 1 k - 1 ) . That is, explain why the right-hand side counts the number of ways to select k elements from a set of n . (b) (10 points) How many sequences are there of the integers { 1 , . . . , n } such that 1 appears before 2? (123 . . . n and 345 . . . n 12 are examples.) You may leave your answer in combination or permutation form, but your answer must be in closed form (i.e., no summations).
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2. (20 points) Probability. Let n be an even integer. Suppose you have a sack with n balls numbered { 1 , . . . , n } . Compute the following. You may leave your answer in combination or permutation form, but your answer must be in closed form (i.e., no summations).
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