4203hw2 - A, B [ n ] so that A B n = ? In that problem we...

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Combinatorics (MAD 4203) — Fall 2010 Due Wednesday 9/22/2010 Homework #2 1 A school has 105 students, and seven classes. If each student takes three classes, and each class is taken by the same number of students, how many students are taking each class? 2 How many northeastern lattice paths 1 are there from (0 , 0) to (10 , 10) which touch the point (5 , 5) but do not touch the point (3 , 3)? 3 Prove that for all positive integers n , p 2 n n P = n s k =0 p n k P 2 . Hint: p n k P = p n n k P . 4 On Homework #1 we had the following problem. 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. Exactly how many n -element subsets of [2 n ] are there in which no two elements diFer by 1? 5 On Homework #1 we had the following problem. In how many ways can you select two subsets
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Unformatted text preview: A, B [ n ] so that A B n = ? In that problem we considered the sets A and B to be diFerent, so for example, we counted both pairs of sets A = { 1 , 4 , 6 } , B = { 2 , 5 , 6 } and A = { 2 , 5 , 6 } , B = { 1 , 4 , 6 } . The answer was 4 n 3 n . This time, dont make this distinction, and instead answer the following question. In how many ways can you select two subsets of [ n ] which intersect? 1 A northeastern lattice path is a walk on the grid of points with integer coordinates that uses only the steps (1 , 0) and (0 , 1). For example, there are 10 northeastern lattice paths from (0 , 0) to (2 , 3):...
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This note was uploaded on 12/10/2011 for the course MAD 4203 taught by Professor Vetter during the Fall '10 term at University of Florida.

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