---
tag: 0WFCGVMRRC
kind: definition
name: Factorial
msc: [05A10]
defines: 0XWG0JGDFW
sources:
  - url: https://en.wikipedia.org/wiki/Factorial
    locator: Definition
---

## Definition

Let $n$ be a natural number.
The factorial of $n$ is the natural number $n!$ defined by the recursion
$$0! = 1, \qquad (n+1)! = (n+1) \cdot n!.$$
For $n \ge 1$ this gives $n! = 1 \cdot 2 \cdots n$, the product of the positive integers up to $n$.

## Remarks

The first values are $0! = 1$, $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$ and $5! = 120$.
The value $n!$ is the [[number of permutations|0CEBCS0E0E]] of an $n$-element set.
The convention $0! = 1$ agrees with the empty product, and it makes the [[factorial formula for binomial coefficients|033KTKKRV6]] true for $k = 0$ and $k = n$.

## Explanations

### L1 human

The factorial $n!$ multiplies all whole numbers from $1$ to $n$.
It counts the ways to put $n$ different books in a row: $n$ choices for the first place, $n - 1$ for the second, and so on.