---
tag: 04JHQNNBFE
kind: statement
role: theorem
name: Hockey-stick identity
msc: [05A19]
uses: [092GEF38YD]
sources:
  - url: https://en.wikipedia.org/wiki/Hockey-stick_identity
    locator: Formulations
same_as:
  wikidata: Q28403965
  wikipedia: 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|05DME4TZ3K]] 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|05DME4TZ3K]] 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.