Math 239 Assignment 1

# Math 239 Assignment 1 - the right. Enumerate the number of...

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MATH 239 Assignment 1 DUE: NOON Friday 14 January 2011 in the drop boxes opposite the Math Tutorial Centre MC 4067. 1. Let k and be non-negative integers such that k . Give two proofs, one combinatorial and one using the binomial theorem, of k X i =0 ± m i ²± n k - i ² = ± m + n k ² . (Hint. For the combinatorial proof, what does the right-hand side count? How can we break up these objects into disjoint groups that are (separately) counted by the factors on the left-hand side?) 2. Consider the standard chessboard, which is 8 squares by 8 squares. Label these squares so the bottom left corner is (0 , 0), the top left corner is (0 , 7), the top right corner is (7 , 7), and the bottom right corner is (7 , 0) (that is, in a natural xy -fashion). Our travels allow us to move in only two directions from a given square: in a single step we may move either 1 square upwards or 1 square to
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Unformatted text preview: the right. Enumerate the number of ways to get by such steps from: (a) (0 , 0) to (1 , 2); (b) (0 , 0) to (2 , 3); (c) (0 , 0) to (4 , 2); and (d) (0 , 0) to (7 , 7). 3. For the chessboard labelled as in the preceding question, let a ( x,y ) denote the number of ways to get from (0 , 0) to the square labelled ( x,y ). Suppose x > and y > 0. Give a combinatorial proof that a ( x,y ) = a ( x-1 ,y ) + a ( x,y-1). 4. Let S = { 1 , 2 , 3 } . Below we describe various weight functions w for each subset T of S . In each case, write down the corresonding generating function. (a) w ( T ) is | T | (that is, the number of elements of T . (b) w ( T ) is the smallest number in T (if T = , then w ( T ) = 4). (c) w ( T ) is the sum of the elements of T (if T = , then w ( T ) = 0)....
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## This note was uploaded on 06/12/2011 for the course CO 250 taught by Professor Guenin during the Spring '10 term at Waterloo.

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