Quiz 1 - Spring002602Q1 .nb http://wwwmath.

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Unformatted text preview: Spring002602Q1 .nb http://wwwmath. gatech.edu/~morley/2602/quizes/Q1/index .html Math 2602 Jan 25,2000 Name: Section -- Circle One: Alan Michaels Shuli Fu Lisa McShine Don 't Panic BTW: All of the people mentioned in the problems are mountain climbers. All but Georgie Mallory have climed Everest. (Malory almost certainly did not, he died trying.) Open Book and Notes. You have 50 minutes. Carefully explain your proceedures and answers Ml answers should he g1m exactly {no decimals} 1f possible . Problem 1 (10 points) George Mallory wants you to show by induction that R? + n is evenly divisibleby 2, for all n ≥ 1 Answer We first check that R2 + n is evenly divisibleby 2 for n = 1. But 1 2 + 1 = 2, which is diviible by 2. Now assume that n2 + n is divisible by 2. And Look at m+132 +(n+1)=n2+2n +1+n+1=n2+n +2n+2 By induction 3%: + n is divible by 2. 2n + 2 is divisible by 2, therefore the sum is divisible by 2, and the result follows. Problem 2 (10 Points) Edmond Hillary want you to show (you need not use induction) that n! ≤ R" for all n ≥ I of 2 09/03/2003 11:46 PM Spfing002602Q1.nb http:l/www.math.gatech.edu/~morley/2602/quizes/Q1/index.html 1. Answer n!=1254bb n nn=nnnnnn .11 Note that each term inh‘1 is n, 1where each term 111 n! 13 s n. The result follows. Problem 3 (10 points) Lopsang J angbu Sherpa needs to know: In a committe of size n ≥2, how many way are there to create two subcommitees A and B, each with a designated member called the chair,such that each person can be on 0,1,or 2 subcommittees,but the chair of subcommitte A cannot be the member of subcommitee B and the chair of committee B cannot be a memeber of committee A? Ans Pick the chair of committee A, there are n ways to do this. Pich the chian' of committe B, there are n—l ways to do this. There are (n-2) people left weach ofwhich can serve on none, A, B, or AB. Therefore there are 4 choices. Answer is ...... .. n(n-1) 4""2 Problem 4. (10 points) Anatoli Boukreev wants to know how many ways are there to get 7 cards (out of 52) with 3 cards of one rank, 2 cards of another rank, the rest of the cards unrelated (xxxyyzwr). Answer Note : Blnmual [11, k] 15 the hmmual coefficient n choose 1:: |n[5]:= 13 Emanuel [4. 3] 12 Burnout-11H, 2] Binomial [11, 2] 42 om]: szya‘rzu Converted by Mathematica January 25, 2000 7. of 2 09/03/2003 11:46 PM ...
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Quiz 1 - Spring002602Q1 .nb http://wwwmath.

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