A little help with a binomial theorem proof

[bennyska]

Homework Statement

(here, (n,k) reads n choose k)
prove that (n,0) - (n, 1) + ... + (-1)ⁿ(n,n) = 0

Homework Equations

binomial theorem

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!

 

[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 $$1^i = 1,\;\forall i.$$

 

[bennyska]

<slaps forehead>
this is exactly the same as the proof i did before, except for the different x and y values.
thanks.