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

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

Factorial can be used to find out how many possible ways there are to arrange 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. That would 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 have been placed. In other words, 3 × 2 × 1 = 6 choices.

The factorial function is a good example of recursion (doing things over and over), as 3! can be written as 3×(2!), which can be written as 3×2×(1!) and finally as 3×2×1×(0!). N! can therefore also be defined as N×(N-1)!,[4] with 0! = 1.[3]

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


n! is not defined for negative numbers. However, the related gamma function is defined over the real and complex numbers (but the integers it is defined over are positive).[3]

Related pagesEdit


  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. 3.0 3.1 3.2 3.3 Weisstein, Eric W. "Factorial". mathworld.wolfram.com. Retrieved 2020-09-09.
  4. 4.0 4.1 "Factorial Function !". www.mathsisfun.com. Retrieved 2020-09-09.

Other websitesEdit