What Is a Factorial and How Is It Used?
The factorial of a non-negative integer n (written n!) is the product of all positive integers from 1 to n. Factorials grow extraordinarily quickly — 20! exceeds 2 quintillion. They appear throughout combinatorics, probability, and calculus, most notably in permutation and combination formulas that count the ways to arrange or select items.
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Formula
Factorial Calculator
Calculate the factorial of any non-negative integer (n!).
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Worked Example
Given:
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FAQs
Why does 0! = 1?
By convention, 0! = 1. This is defined to make combination and permutation formulas work correctly when n = 0. It is also consistent with the recursive definition n! = n × (n-1)!: if 1! = 1, then 0! = 1!/1 = 1.
Where are factorials used in everyday mathematics?
Factorials appear in permutations (number of ways to arrange n items = n!) and combinations (n choose k = n! / (k!(n-k)!)). For example, the number of ways to arrange 5 books on a shelf is 5! = 120. The number of possible 5-card poker hands uses combination formulas.
Why can't calculators handle very large factorials?
Factorials grow faster than exponential functions. 170! ≈ 7.26 × 10³⁰⁶ is near the maximum value a standard floating-point number can represent. Beyond 170!, the result exceeds the range of standard double precision numbers, requiring arbitrary-precision arithmetic libraries.