We can always view $\binom{x}{k}$ as a polynomial in $x$ of degree $k$. With this in mind, why is it so that a polynomial $f\in\mathbb{Q}[x]$ is such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$ iff the coefficients of $f$ in terms of the basis $\{\binom{x}{k}\mid k\in\mathbb{N}\}$ are also integers?
I thought it might be useful to note that $0,1,\dots,k-1$ are roots of $\binom{x}{k}$, but I still don't see why such a property would be true. Thanks for an explanation.