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mt2-sol

# mt2-sol - Math 55-2 Midterm 2 SOLUTIONS Rob Bayer 1(10 pts...

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Math 55-2 Midterm 2 SOLUTIONS Rob Bayer July 27, 2009 1. (10 pts) Short answer. You need not show any work for this section (a) How many ways are there to put n balls in k bins if: i. The balls and the bins are both numbered k n ii. The balls are indistinguishable and the bins are numbered n + k - 1 k - 1 iii. The balls are numbered and the bins are indistinguishable k X i =1 S ( n, i ) iv. Both the balls and the bins are indistinguishable p k ( n ) (b) Find the coefficient of x 8 in (5 x - 7) 21 ( 21 8 ) 5 8 ( - 7) 13 (c) State the Generalized Pigeonhole Principle: If you put n pigeons in k boxes, some box contains at least d n k e pigeons (d) Define what it means for a permutation to be a derangement: The permutation has no fixed points. (ie, every element gets moved) (e) Find the flaw in the following “proof” that if a 6 = 0, then a n = 1 for all non-negative integers n : BC (n=0) : a 0 = 1 since a 6 = 0 IH : Suppose the claim is true for all exponents k . (Formally: a i = 1 for all i k ) IS : a k +1 = a k · a k a k - 1 IH = 1 · 1 1 = 1 There aren’t enough base cases–when trying to prove it for a 1 (ie, k = 0) we need an a - 1 which we don’t have a base case for. (In general, since we go down 2 steps in our IS, we need 2 base cases) 2. (15 pts) Prove that 1 1 · 3 + 1 3 · 5 + 1 5 · 7 + · · · + 1 (2 n - 1)(2 n + 1) = n 2 n + 1 for every n 1 We’ll go by induction on n : BC (n=1) The LHS is just 1 3 and the RHS is 1 2+1 = 1 3 IH Suppose that 1 1 · 3 + 1 3 · 5 + 1 5 · 7 + · · · + 1 (2 k - 1)(2 k +1) = k 2 k +1 IS 1

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1 1 · 3 + 1 3 · 5 + 1 5 · 7 + · · · + 1 (2 n - 1)(2 n + 1) + 1 (2 k + 1)(2 k + 3) IH = k 2 k + 1 + 1 (2 k + 1)(2 k + 3) = k (2 k + 3) + 1 (2 k + 1)(2 k + 3) = 2 k 2 + 3 k + 1 (2 k + 1)(2 k + 3) = ( k + 1)(2 k + 1) (2 k + 1)(2 k + 3) = k + 1 2 k + 3 Thus, by induction, the result holds for all n 1 3. (15 pts) Consider the set S defined as follows: 001 S Whenever w, v S , 100 w S, 00 w 1 S and wv S Prove that every string in S ends with a 1 and contains twice as many 0’s as 1’s.
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mt2-sol - Math 55-2 Midterm 2 SOLUTIONS Rob Bayer 1(10 pts...

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