Calculate factorials with step-by-step solutions, scientific notation for large values, and applications in permutations and combinations.
Factorial
120
Enter a non-negative integer (0-170)
5! = 120
(3 digits)
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%
Factorial
120
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.
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.
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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.
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.
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.
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.
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: 2025-01-15
Factorial
120