Math 2602

Jan 25,2000


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


Problem 1 (10 points)

George Mallory  wants you to show by induction that [Graphics:Images/index_gr_2.gif] + n  is evenly divisibleby 2, for all n ≥ 1


We first check that [Graphics:Images/index_gr_3.gif] + n  is evenly divisibleby 2 for n = 1. But  1[Graphics:Images/index_gr_4.gif] + 1 = 2, which is diviible by 2.

Now assume that  [Graphics:Images/index_gr_5.gif] + n is divisible by 2. And Look at

[Graphics:Images/index_gr_6.gif] + (n+1) = [Graphics:Images/index_gr_7.gif] + 2 n  + 1 + n + 1 = [Graphics:Images/index_gr_8.gif] + n  + 2n + 2

By induction [Graphics:Images/index_gr_9.gif] + 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! ≤ [Graphics:Images/index_gr_11.gif] for all n ≥ 1.



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?


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) [Graphics:Images/index_gr_13.gif]

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).



Converted by Mathematica      January 25, 2000