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Let $n \in \Z_{\ge 0}$ be a positive integer.

Definition 1

The factorial of $n$ is defined inductively as:

$n! = \begin {cases} 1 & : n = 0 \\ n \paren {n - 1}! & : n > 0 \end {cases}$

Definition 2

The factorial of $n$ is defined as:

\(\ds n!\) \(=\) \(\ds \prod_{k \mathop = 1}^n k\)
\(\ds \) \(=\) \(\ds 1 \times 2 \times \cdots \times \paren {n - 1} \times n\)

where $\ds \prod$ denotes product notation.


The factorials of the first few positive integers are as follows:

$\begin{array}{r|r} n & n! \\ \hline 0 & 1 \\ 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5 \, 040 \\ 8 & 40 \, 320 \\ 9 & 362 \, 880 \\ 10 & 3 \, 628 \, 800 \\ \end{array}$

Definition for Multiindices

Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \ge 0$.

Then we define:

$\ds \alpha! = \prod_{j \mathop \in J} \alpha_j!$

where the factorial on the right is a factorial of natural numbers.

Note that by definition, all by finitely many of the $\alpha_j$ are zero, so the product over $J$ is convergent.

Also known as

While the canonical vocalisation of $n!$ is $n$ factorial, it can often be found referred to as $n$ bang or (usually by schoolchildren) $n$ shriek.

Some mathematicians prefer $n$ gosh.

Some early sources favour factorial $n$

Also see

  • Results about factorials can be found here.

Historical Note

The symbol $!$ used on $\mathsf{Pr} \infty \mathsf{fWiki}$ for the factorial, which is now universal, was introduced by Christian Kramp in his $1808$ work Élémens d'arithmétique universelle.

Before that, various symbols were used whose existence is now of less importance.

Notations for $n!$ in history include the following:

$\sqbrk n$ as used by Euler
$\mathop{\Pi} n$ as used by Gauss
$\left\lvert {\kern-1pt \underline n} \right.$ and $\left. {\underline n \kern-1pt} \right\rvert$, once popular in England and Italy.

In fact, Henry Ernest Dudeney was using $\left\lvert {\kern-1pt \underline n} \right.$ as recently as the $1920$s.

Augustus De Morgan declared his reservations about Kramp's notation thus:

Amongst the worst barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation $n!$ ... which gives their pages the appearance of expressing admiration that $2$, $3$, $4$, etc., should be found in mathematical results.
-- 1929: Florian Cajori: A History of Mathematical Notations: Volume $\text { 2 }$

The use of $n!$ for non-integer $n$ is uncommon, as the Gamma function tends to be used instead.