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Unformatted text preview: Problem 1: Give a combinatorial proof of the identity
n+m
k Problem 2: k m
i =
i=0 n
k−i Give a combinatorial proof of the identity
n 3n =
k=0 nk
2.
k Problem 3: Let an,k be the number of subsets of {1, . . . , n} of size k that
contains no consecutive pairs (i.e. i and i + 1 form a consecutive pair for any
i = 1, . . . , n − 1). Give a combinatorial proof that
an,k =
Problem 4: n−k+1
.
k Give a combinatorial proof to the identity k ≥0 n
2k =
k≥0 n
.
2k + 1 Problem 5: Let S be the set of all subset of {1, 2, 4, 6, 7} of size two . Deﬁne
w(A) = (a − b)2 = (b − a)2 where A = {a, b} ∈ S . Write the generating function
of S with respect to this weight. 1 ...
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This note was uploaded on 11/26/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
 Spring '09
 M.PEI
 Math, Sets

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