The Helpful Mathematics
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Article 11 — Appendix A.21

n! — factorial

Category. Mathematics.

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

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

See also. Factorial generalization — gamma function Γ.

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]!;