Factorial¶

Definition¶

Let $n$ be a natural number. The factorial of $n$ , denoted $n!$ , is the product of all positive natural numbers up to $n$ :

n!=1\times 2\times 3\times \cdots\times (n-1)\times n.

In addition, $0!$ is defined to be $1$ .

Interpretation¶

The factorial counts how many ways distinct objects can be arranged in order. For $n$ objects there are $n$ choices for the first position, then $n-1$ for the second, and so on down to $1$ for the last; the choices multiply, giving $n!$ .

The definition can also be read recursively: $0!=1$ and $(n+1)!=(n+1)\cdot n!$ $n!$ , the form used in proofs by mathematical induction. The convention $0!=1$ fits this recursion, since $1!=1\cdot0!$ $0!$ , and it fits the counting meaning, since there is exactly one way to arrange no objects.

Factorials grow very fast: each step multiplies by $n+1$ $1$ , an ever larger factor, whereas a power $a^n$ multiplies by the fixed factor $a$ . A common slip is to treat the factorial as if it were linear or multiplicative: $(2n)!$ is neither $2\cdot n!$ nor $(n!)^2$ . For $n=3$ $3$ , $6!=720$ $720$ , while $2\cdot3!=12$ and $(3!)^2=36$ $36$ .

In probability, factorials return whenever outcomes are counted through ordered arrangements or selections.

Key Results¶

There are $n!$ permutations of $n$ distinct objects.
Factorials appear in binomial coefficients:

\binom{n}{k}=\frac{n!}{k!(n-k)!}.

The factorial function grows faster than any fixed polynomial or exponential function.

Examples¶

$4!=24$ $24$ .
$10!=3628800$ $3628800$ .
Three different apples can be assigned to three ordered people in $3!=6$ ways.
A full ordering of 52 distinct cards has $52!$ possible orders.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 1
Probability Theory for EOR · Week 1

Aliases: n factorial

Last updated: 20 October 2026 03:02