Proof with binomial coefficient and kronecker delta

Proof with binomial coefficient and kronecker delta

I want to prove that $$ \sum_{k=i}^n
\binom{n}{k}\binom{k}{i}(-1)^{n-k}=\delta_{n,i} $$ Where $\delta_{n,i}$ is
the Kronecker Delta, i.e. $\delta_{n,i}=0$ if $n \neq i$ and
$\delta_{n,i}=1$ if $i=n$.
I have an intuitive proof with matrices representing the linear
transformation $F(x)=(1+x)^n$ and $G(x)=(1-x)^n$ and their product. But
I'd prefer an accurate proof, if it exists. Thanks.