Factorial Calculator
Calculate factorials (n!), double factorials (n!!), subfactorials (!n), and Stirling's approximation
Calculate Factorial Functions
Enter any non-negative integer (0 to 170 for regular factorial)
Choose the type of factorial calculation
Function Definitions:
Common Factorial Values
Factorial Tips
0! = 1 by mathematical convention
Factorials grow extremely quickly
Used in permutations and combinations
Double factorials skip every other number
Subfactorials count derangements
Stirling's formula is very accurate for large numbers
Understanding Factorials and Related Functions
What is a Factorial?
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. It's fundamental in combinatorics, probability, and many areas of mathematics.
Mathematical Definition:
n! = n × (n-1) × (n-2) × ... × 2 × 1
For n ≥ 1, and 0! = 1 by definition
Why is 0! = 1?
This is defined by mathematical convention to make formulas work consistently. It ensures that the recursive formula n! = n × (n-1)! holds for all positive integers.
Applications in Mathematics
Permutations
Number of ways to arrange n objects: P(n,r) = n!/(n-r)!
Combinations
Number of ways to choose r objects from n: C(n,r) = n!/(r!(n-r)!)
Probability Theory
Calculating probabilities in discrete distributions
Calculus
Taylor series expansions and power series
Factorial Variations
Regular Factorial (n!)
Standard factorial function
5! = 5 × 4 × 3 × 2 × 1 = 120
Double Factorial (n!!)
Product with same parity numbers
5!! = 5 × 3 × 1 = 15
Subfactorial (!n)
Number of derangements
!4 = 9
Stirling's Approximation
For large values of n, calculating n! exactly becomes computationally intensive. Stirling's approximation provides a very accurate estimate:
n! ≈ √(2πn) × (n/e)ⁿ
This approximation becomes increasingly accurate as n grows larger, with the relative error approaching zero.