# Factorial

product of all integers between 1 and the integral input of the function

The factorial of a whole number ${\displaystyle n}$, written as ${\displaystyle n!}$[1] or ${\displaystyle n}$,[2] is found by multiplying ${\displaystyle n}$ by all the whole numbers less than it. For example, the factorial of 4 is 24, because ${\displaystyle 4\times 3\times 2\times 1=24}$. Hence one can write ${\displaystyle 4!=24}$. For some technical reasons, ${\displaystyle 0!}$ is equal to ${\displaystyle 1}$.[3]

Factorials can be used to find out how many possible ways there are to arrange ${\displaystyle n}$ objects.[3]

For example, if there are 3 letters (A, B, and C), they can be arranged as ABC, ACB, BAC, BCA, CAB, and CBA. This results in 6 choices, as A can be put in 3 different places, B has 2 choices left after A is placed, and C has only one choice left after A and B are placed. In other words, ${\displaystyle 3\times 2\times 1=6}$ choices.

The factorial function is a good example of recursion (doing things over and over), as ${\displaystyle 3!}$ can be written as ${\displaystyle 3\times 2!}$, which can be written as ${\displaystyle 3\times 2\times 1!}$ and finally as ${\displaystyle 3\times 2\times 1\times 0!}$. Because of this, ${\displaystyle n!}$ can also be defined as ${\displaystyle n\times (n-1)!}$,[4] with ${\displaystyle 0!=1}$[3]

The factorial function grows very fast. There are ${\displaystyle 10!=3,628,800}$ ways to arrange 10 items.[4]

## Applications

The earliest uses of the factorial function involve counting permutations: there are ${\displaystyle n!}$  different ways of arranging ${\displaystyle n}$  distinct objects into a sequence.[5] Factorials appear more broadly in many formulas in combinatorics, to account for different orderings of objects. For instance the binomial coefficients ${\displaystyle {\tbinom {n}{k}}}$  count the ${\displaystyle k}$ -element combinations (subsets of ${\displaystyle k}$  elements) from a set with ${\displaystyle n}$  elements, and can be computed from factorials using the formula[6]

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

## Related sequences and functions

Several other integer sequences are similar to or related to the factorials:

Alternating factorial
The alternating factorial is the absolute value of the alternating sum of the first ${\displaystyle n}$  factorials, ${\textstyle \sum _{i=1}^{n}(-1)^{n-i}i!}$ . These have mainly been studied in connection with their primality; only finitely many of them can be prime, but a complete list of primes of this form is not known.[7]

## Notes

Factorials are not defined for negative integers. However, the related gamma function (${\displaystyle \Gamma (x)}$ ) is defined over the real and complex numbers (except for negative integers).[3]

## References

1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-09-09.
2. Aggarwal, M.L. (2021). "8. Permutations and Combinations". Understanding ISC Mathematics Class XI. Vol. I. Industrial Area, Trilokpur Road, Kala Amb-173030, Distt. Simour (H.P.): Arya Publications (Avichal Publishing Company). p. A-400. ISBN 978-81-7855-743-4.{{cite book}}: CS1 maint: location (link)
3. Weisstein, Eric W. "Factorial". mathworld.wolfram.com. Retrieved 2020-09-09.
4. "Factorial Function !". www.mathsisfun.com. Retrieved 2020-09-09.
5. Conway, John H.; Guy, Richard (1998). "Factorial numbers". The Book of Numbers. Springer Science & Business Media. pp. 55–56. ISBN 978-0-387-97993-9.
6. Graham, Knuth & Patashnik 1988, p. 156.
7. Guy 2004. "B43: Alternating sums of factorials". pp. 152–153.