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# Factorial

The **factorial** of a whole number *n,* written as **n!**,* ^{[1]}* 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*.

^{[2]}

Factorial can be used to find out how many possible ways there are to arrange n objects.^{[2]}

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)!,^{[3]} with 0! = 1.^{[2]}

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

## Notes

*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).^{[2]}

## Related pages

## References

- ↑ "Compendium of Mathematical Symbols" (in en-US). 2020-03-01. https://mathvault.ca/hub/higher-math/math-symbols/.
- ↑
^{2.0}^{2.1}^{2.2}^{2.3}Weisstein, Eric W.. "Factorial" (in en). https://mathworld.wolfram.com/Factorial.html. - ↑
^{3.0}^{3.1}"Factorial Function !". https://www.mathsisfun.com/numbers/factorial.html.