Hockey-stick identity

Statement

Let $n$ and $r$ be natural numbers with $r \le n$. Then $$\sum_{i=r}^{n} \binom{i}{r} = \binom{n + 1}{r + 1}.$$

Remarks

For $r = 1$ it gives $1 + 2 + \dots + n = \binom{n+1}{2} = \frac{n(n+1)}{2}$. Combinatorial proof: classify the $(r+1)$-subsets of $\{1, \dots, n + 1\}$ by their largest element $i + 1$. The other $r$ elements come from $\{1, \dots, i\}$. In Pascal's triangle the summands lie on a diagonal, and the result lies below the end of the diagonal. This shape is the reason for the name.

Explanations

L1 human

Add the entries of Pascal's triangle along a diagonal, from the edge downward. The total is the entry just below and to the side of the last one, so the picture looks like a hockey stick.