Why does $\binom{n}{k} = 0$, if $k > n$? I have came across this in a textbook that I am currently studying, but I don't understand how I should proof this.
A short explanation or proof would be nice.

Since
$$
\binom{n}{k}=\prod_{j=0}^{k-1}\frac{n-j}{j+1}\tag1
$$
if $k\gt n\ge0$ and $n\in\mathbb{Z}$, then the product contains a factor of $0$.

In the Generalized Binomial Theorem, $(1)$ is the definition of the binomial coefficients (with the convention that when $k=0$, $(1)$ is an empty product, and therefore, $1$). They are only defined for $k\in\mathbb{Z}$ and for $k\lt0$, $\binom{n}{k}=0$.
However, if $n\not\in\mathbb{Z}$, then $\binom{n}{k}\ne0$ for any integer $k\ge0$.
