# Math 2602

### 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.)

### Problem 1 (10 points)

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

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.

### Problem  2 (10 Points)

Edmond Hillary want you to  show  (you need not use induction) that n! ≤ 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?

#### 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.