Unformatted text preview: na pairs in Ω with ﬁrst element equal to A . Now use the binomial theorem. (b) Describe a bijection between the elements of Ω and the sequences of length n with elements from {0,1,2}. 4. (6 points) Let N ≥ denote the set of nonnegative integers. (a) If the weight of the integer i is i if 3 divides i and i1 otherwise, what is the generating function Φ N ≥ ( x )? (b) Compute the generating function if the weight of i is i , i1 or i2 according to whether i is congruent modulo 3 to 0, 1 or 2. 5. (7 points) If n , k ≥ 1 then ˆ n k ! = ˆ n1 k ! + ˆ n1 k1 ! . Prove this in two ways: (a) The left side is the number of kelement subsets of a set of size n . Explain how we can interpret each of the terms on the right as numbers of subsets, and hence deduce the result. (b) By using the binomial theorem to expand each side of the equation (1 + x ) n = (1 + x )(1 + x ) n1 ....
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 Spring '09
 M.PEI
 Math, Binomial Theorem, appropriate Drop Box, drop box

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