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Math 2602

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Jan 25,2000*

Name:____________________________

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Section -- Circle One:*

Alan Michaels

Shuli Fu

Lisa McShine

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Don't Panic
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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.)

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Open Book and Notes. You have 50 minutes. Carefully explain your proceedures and answers

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Problem 1 (10 points)

George Mallory wants you to show by induction that + n is evenly divisibleby 2, for all n ≥ 1

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Answer

We first check that + n is evenly divisibleby 2 for n = 1. But 1 + 1 = 2, which is diviible by 2.

Now assume that + n is divisible by 2. And Look at

+ (n+1) = + 2 n + 1 + n + 1 = + n + 2n + 2

By induction + n is divible by 2. 2n + 2 is divisible by 2, therefore

the sum is divisible by 2, and the result follows.

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Problem 2 (10 Points)

Edmond Hillary want you to show (you need not use induction) that n! ≤ for all n ≥ 1.

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Answer

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Problem 3 (10 points)

Lopsang Jangbu 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?

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Ans

Pick the chair of committee A, there are n ways to do this. Pich the chiari of committe B, there are n-1 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)

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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).

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Answer

Converted by *Mathematica*
January 25, 2000