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Course: MATH 239, Fall 2009
School: W. Alabama
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1: Consider Problem the identity n k = n-1 n-1 + k k-1 a. Give an algebraic proof for the identity. b. Give a combinatorial proof for the identity. Problem 2: Give a combinatorial proof of the identity n+m k k = i=0 m i n k-i Problem 3: Let S = {1, 2, 3, . . .} and w(i) = i if i is divisible by 3; w(i) = i-1 i if = 1 mod 3 and w(i) = i - 2 if i = 2 mod 3. Write the generating function of S with respect to...

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1: Consider Problem the identity n k = n-1 n-1 + k k-1 a. Give an algebraic proof for the identity. b. Give a combinatorial proof for the identity. Problem 2: Give a combinatorial proof of the identity n+m k k = i=0 m i n k-i Problem 3: Let S = {1, 2, 3, . . .} and w(i) = i if i is divisible by 3; w(i) = i-1 i if = 1 mod 3 and w(i) = i - 2 if i = 2 mod 3. Write the generating function of S with respect to this weight. Problem 4: Let S be the set of all subset of {1, 2, 4, 6, 7} of size two...

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