Calculate Factorial
Enter a non-negative integer (0-170)
5! = 120
(3 digits)
Step-by-Step Solution
Formula: n! = n x (n-1) x (n-2) x ... x 2 x 1
Special case: 0! = 1 (by definition)
5! = 5 x 4 x 3 x 2 x 1
5 x 4 = 20
20 x 3 = 60
60 x 2 = 120
120 x 1 = 120
Result: 5! = 120
Double Factorial
5!! = 15
5 x 3 x ... = 15
Subfactorial (Derangements)
!5 = 44
Number of permutations with no fixed points
Stirling's Approximation
118
Error: 1.6507%
?How Do You Calculate a Factorial?
Factorial of n (written n!) is the product of all positive integers from 1 to n: n! = n x (n-1) x (n-2) x ... x 2 x 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. Special case: 0! = 1 by definition. Factorials grow extremely fast and are used in permutations, combinations, probability, and Taylor series.
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Factorial
120
What is a Factorial?
The factorial of a non-negative integer n, denoted n!, is the product of all positive integers less than or equal to n. Factorials are fundamental in combinatorics for counting permutations and combinations, in probability theory, in calculus for Taylor series expansions, and in many areas of mathematics and science. The factorial function grows faster than exponential functions.
Key Facts About Factorials
- n! = n x (n-1) x (n-2) x ... x 2 x 1
- 5! = 5 x 4 x 3 x 2 x 1 = 120
- 0! = 1 (by definition)
- Factorials grow very fast: 10! = 3,628,800
- Used in permutations: P(n,r) = n!/(n-r)!
- Used in combinations: C(n,r) = n!/(r!(n-r)!)
- 20! has 19 digits, 100! has 158 digits
- Stirling approximation: n! is approximately sqrt(2 x pi x n)(n/e)^n
Frequently Asked Questions
What is a factorial?
The factorial of n (written n!) is the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. By convention, 0! = 1. Factorials count the number of ways to arrange n distinct objects.
Why is 0! equal to 1?
There are two reasons: (1) Combinatorially, there is exactly one way to arrange zero objects - do nothing. (2) Mathematically, the recurrence n! = n x (n-1)! requires 0! = 1 for 1! = 1 x 0! to equal 1.
How fast do factorials grow?
Factorials grow faster than any exponential function. 10! = 3,628,800 (7 digits), 20! = 2.4 x 10^18 (19 digits), 100! has 158 digits. By Stirling's approximation, n! is approximately sqrt(2*pi*n) x (n/e)^n.
What is a double factorial?
The double factorial n!! multiplies every other number: n!! = n x (n-2) x (n-4) x ... down to 1 or 2. For example: 7!! = 7 x 5 x 3 x 1 = 105, and 8!! = 8 x 6 x 4 x 2 = 384.
What is the subfactorial?
The subfactorial !n counts derangements - permutations where no element is in its original position. For example, !3 = 2: the arrangements (2,3,1) and (3,1,2) are the only derangements of (1,2,3). Formula: !n = n! x sum of (-1)^k/k! from k=0 to n.
Last updated: 2026-01-15
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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