4203hw1 - n couples to a party. She wants to ask a...

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Combinatorics (MAD 4203) — Fall 2010 Due Wednesday 9/8/2010 Homework #1 1 Prove that if n + 1 numbers are chosen from the set { 1 , 2 , 3 , 4 , . . . , 2 n } then there must be two which diFer by 1. 2 Prove that of any ±ve points chosen within an equilateral triangle of side length 1, there are two whose distance apart is at most 1 / 2 . 3 Let r be any irrational number. Prove that there exists a positive integer n so that nr lies within 10 - 10 of an integer. 4 Prove that for all positive integers n , 1 3 + 2 3 + · · · + n 3 = (1 + 2 + · · · n ) 2 . 5 In how many ways can the elements of [ n ] be permuted if 1 must precede 2 and 3? 6 In how many ways can you select two subsets A, B [ n ] so that A B n = ? 7 A host invites
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Unformatted text preview: n couples to a party. She wants to ask a (possibly empty) subset of the 2 n guests to give a speech, but she does not want to ask both members of any couple to give speeches. In how many ways can she do this? Answers to the following questions will help me with the direction of the course, in addition to satisfying my curiosity. 8 Please describe your history with mathematics. Possible angles in-clude: What classes have you enjoyed the most? What problems have interested you? How did you end up studying mathematics? 9 What are your future plans for mathematics? What are you studying here?...
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