TITLE: Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
QUESTION [14 upvotes]: 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?

REPLY [5 votes]: How many solutions are there to $a_1+\cdots+a_k\leq n$ in nonnegative integers?
On the one hand, this is
$$\begin{align*}&\sum_{i=0}^n\#\{(a_1,\ldots,a_k)\mid a_1+\cdots+a_k=i\}\\
=& \sum_{i=0}^n\binom{i+k-1}{k-1}\end{align*}$$
On the other had, every solution to $a_1+\cdots+a_k\leq n$ corresponds to a unique solution of $a_1+\cdots+a_k+t=n$ by setting $t=n-(a_1+\cdots+a_k)$.
Hence $$\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}k.$$