The Art of Interface

# n! — factorial

Category. Mathematics.

Abstract. Factorial: definition, properties, identities and table of values for some arguments.

## 1. Definition

Factorial is defined as

0! ≡ 1;
n! ≡ 1×2×...×(n − 1) × n, for n = 1, 2, 3, ...

From definition follows, that factorial is defined only for non-negative integers, and its value is always positive integer. So, as function it is defined only at discrete points.

## 2. Identities

Next value identity

(n + 1)! = (n + 1) n!

There is a generalization of the factorial for real numbers — gamma function Γ — and

n! = Γ(n + 1)

## 3. Combinatorics

Number of permutations — ordered k-size subsets of n-element set:

nPk = n! /(nk)!

Number of combinations — k-size subsets of n-element set:

nCk = n! / [k! (nk)!]

First dozen values

nn!
01
11
22
36
424
5120
6720
75040
840320
9362880
103628800
1139916800
12479001600
Table. 1. Factorial for some n.

## 4. Support

Factorial n! is supported in:

Factorial of the real number n!=x! (resolved into gamma function) is supported in:

Factorial of the complex number n!=z! (resolved into gamma function of the complex argument) is supported in:

## 5. How to use

To calculate factorial of the number:

``8!;``

To calculate factorial of the current result:

``Rslt!;``

To calculate factorial of the number n in memory:

``Mem[n]!;``