mt2-sol - Math 55-2 Midterm 2 SOLUTIONS Rob Bayer July 27,...

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Unformatted text preview: 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 = 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 arent enough base caseswhen trying to prove it for a 1 (ie, k = 0) we need an a- 1 which we dont 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 Well 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 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 0s as 1s.ends with a 1 and contains twice as many 0s as 1s....
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This note was uploaded on 08/31/2009 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.

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mt2-sol - Math 55-2 Midterm 2 SOLUTIONS Rob Bayer July 27,...

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