Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity) Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity
$$
\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}
$$
holds?

Generating function can do the job quite easily:
\begin{align*}
  \frac{1}{(1-x)^k} &= \sum_{i\ge 0} \binom{i+k-1}{k-1}\, x^i
\end{align*}
Using convolution of generating functions,
\begin{align*}
  \frac{1}{(1-x)}\cdot \frac{1}{(1-x)^k} &= \sum_{n\ge 0} \left(\sum_{i=0}^n \binom{i+k-1}{k-1}\right) x^n \\
   \frac{1}{(1-x)^{k+1}} &=  \sum_{n\ge 0} \binom{n+k}{k} x^n \\
\implies \sum_{i=0}^n \binom{i+k-1}{k-1} &= \binom{n+k}{k}
\end{align*}
