Factorial

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 of an $n$-element set. The convention $0! = 1$ agrees with the empty product, and it makes the factorial formula for binomial coefficients 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.