Factorial
The factorial of a whole number , written as [1] or ,[2] is found by multiplying by all the whole numbers less than it. For example, the factorial of 4 is 24, because . Hence one can write . For some technical reasons, 0! is equal to 1.[3]
Factorials can be used to find out how many possible ways there are to arrange 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. That is be 6 choices because 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, choices.
The factorial function is a good example of recursion (doing things over and over), as can be written as , which can be written as and finally as . Because of this, can also be defined as ,[4] with [3]
The factorial function grows very fast. There are ways to arrange 10 items.[4]
Applications
The earliest uses of the factorial function involve counting permutations: there are different ways of arranging 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 count the -element combinations (subsets of elements) from a set with elements, and can be computed from factorials using the formula[6]
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 factorials, . 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 numbers. However, the related gamma function () is defined over the real and complex numbers (except for negative integers).[3]
Related pages
References
- "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-09-09.
- 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.
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: CS1 maint: location (link) - Weisstein, Eric W. "Factorial". mathworld.wolfram.com. Retrieved 2020-09-09.
- "Factorial Function !". www.mathsisfun.com. Retrieved 2020-09-09.
- 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.
- Graham, Knuth & Patashnik 1988, p. 156.
- Guy 2004 . "B43: Alternating sums of factorials". pp. 152–153.