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hw608sol - CS 1050 B Constructing Proofs April 2 2008...

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CS 1050 B: Constructing Proofs April 2, 2008 Solutions to Homework 6 Lecturer: Sasha Boldyreva Problem 6.1, 20 points. Problem 8 from Section 6.2 of Rosen’s textbook. a) Since 1 has either t precede 2 or follow it, and there is no reason that one of these should be any more likely than the other, we immediately see that the answer is 1/2. b) Similarly, the answer is 1/2. c) For 1 to immediately precede 2, think of the two numbers glued together. Then we are really permuting n-1 numbers – the single numbers from 3 to n and the glued number, 12. There are ( n - 1)! ways to do this. And there are n ! permutations total. Hence the answer is ( n - 1)! /n !. d) Half of the permutations have n preceding 1. Of these permutations, half of them have n-1 preceding 2. Therefore one fourth of the permutations satisfy these conditions, so the probability is 1/4. e) Looking at the relative placements of 1,2, and n, we see that one third of the time, n will come first. Thus the answer is 1/3.
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