# Homework Help: A little help with a binomial theorem proof

1. Jun 23, 2010

### bennyska

1. The problem statement, all variables and given/known data
(here, (n,k) reads n choose k)
prove that (n,0) - (n, 1) + ... + (-1)n(n,n) = 0

2. Relevant equations

binomial theorem

3. The attempt at a solution
so this proof is relatively straightforward when n is odd. it's just matching up terms and having them cancel each other out. i'm having a problem proving it when n is even, because each term doesn't match up exactly. and the middle term also alternates between plus or minus depending on whether n/2 is even. (i think i have the middle term is (-1)n(n,n/2).
but anyway, i've been having trouble with it. a little hint or two would be nice. gracias!

2. Jun 23, 2010

### tmccullough

That's a good approach, but fails for even n, like you noticed.

Consider directly applying the binomial theorem

$$(x+y)^n = \sum_{i=0}^n (n,i)*x^iy^{n-i}$$

Now, just pick the right values for $$x,y.$$

Remember the obvious fact that $$1^i = 1,\;\forall i.$$

3. Jun 24, 2010