---
tag: 033KTKKRV6
kind: statement
role: theorem
name: Factorial formula for binomial coefficients
msc: [05A10]
uses: [0WFCGVMRRC, 092GEF38YD]
sources:
  - url: https://en.wikipedia.org/wiki/Binomial_coefficient
    locator: Factorial formula
---

## Statement

Let $n$ and $k$ be natural numbers with $k \le n$.
Then
$$\binom{n}{k} = \frac{n!}{k!\,(n - k)!}.$$

## Remarks

Equivalently, $k!\,(n-k)!\,\binom{n}{k} = n!$. In this form the identity has no division.
The formula needs $k \le n$. For $k > n$ the left side is $0$ and $(n - k)!$ has no meaning in the natural numbers.
A cancelled form is $\binom{n}{k} = \frac{(n)_k}{k!}$, with the [[falling factorial|0N5M924N3P]] $(n)_k$.

## Explanations

### L1 human

Arrange all $n$ objects in a row in $n!$ ways, and take the first $k$ as the chosen set.
Each choice of set appears $k!\,(n-k)!$ times, because the first $k$ and the last $n-k$ objects can be reordered freely.