---
tag: 092GEF38YD
kind: definition
name: Binomial coefficient
msc: [05A10]
defines: 0GBZM0MT0F
sources:
  - url: https://en.wikipedia.org/wiki/Binomial_coefficient
    locator: Definition and interpretations
---

## Definition

Let $n$ and $k$ be natural numbers.
The binomial coefficient $\binom{n}{k}$, read "$n$ choose $k$", is the natural number defined by the recursion
$$\binom{n}{0} = 1, \qquad \binom{0}{k+1} = 0, \qquad \binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$$

## Remarks

The value $\binom{n}{k}$ is the [[number of k-element subsets|0BFMR2FQ8B]] of an $n$-element set. Many texts take this as the definition.
For $k > n$, $\binom{n}{k} = 0$. For $k \le n$, the [[factorial formula|033KTKKRV6]] gives $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$.
The recursion is [[Pascal's rule|0YRG241HQQ]], and the values form [[Pascal's triangle|05DME4TZ3K]].

## Explanations

### L1 human

The number $\binom{n}{k}$ tells how many ways there are to choose $k$ objects out of $n$ when the order does not matter.
For example, $\binom{4}{2} = 6$: from $\{1, 2, 3, 4\}$ you can choose $12, 13, 14, 23, 24, 34$.