Hello,

let's assume the following set:

$\displaystyle Q_n = \{(x_1, x_2,...,x_n) \in \mathbb{R}^n | x_i \in \{0, 1, 2, 3\} \ \forall i \}$.

Vector $\displaystyle (x_1, x_2,...,x_n)$ in the set $\displaystyle Q_n$ is even, if the integer $\displaystyle x_1+x_2+...+x_n$ is even, else vector is odd. The problem is to show using binomial coefficients, that half of the vectors in set $\displaystyle Q_n$ are even.

It's easy to see that for $\displaystyle n=1$, there's two combinations of four, which are even. Arbitrary $\displaystyle n=k$:

For $\displaystyle x_1+x_2+...+x_{k-1} \ \frac{1}{2}$ are even.

So, with $\displaystyle +x_k$ we get $\displaystyle \frac{2}{4}=\frac{1}{2}$.

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