Hello,

let's assume the following set:

.

Vector in the set is even, if the integer is even, else vector is odd. The problem is to show using binomial coefficients, that half of the vectors in set are even.

It's easy to see that for , there's two combinations of four, which are even. Arbitrary :

For are even.

So, with we get .

Question is, how to do that withbinomial coefficients. Any help is appreciated. Thank you!