🔢 Factorial Calculator

What is a Factorial?

The factorial of a non-negative integer n, written as n!, is the product of all positive integers from 1 to n. For example: 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow incredibly rapidly — 20! exceeds 2 quintillion. Our factorial calculator computes n! instantly for any non-negative integer, even very large ones.

Factorial Formula and Special Cases

n! = n × (n-1) × (n-2) × ... × 2 × 1. Special cases: 0! = 1 (by convention — one way to arrange zero items). 1! = 1. 2! = 2. 3! = 6. 4! = 24. 5! = 120. 10! = 3,628,800. 20! = 2,432,902,008,176,640,000. The recursive definition: n! = n × (n-1)! is foundational in computer science.

Factorials in Permutations and Combinations

Factorials are the foundation of counting problems. Permutations (arrangements where order matters): P(n,r) = n! / (n-r)! — ways to arrange r items from n. Combinations (selections where order does not matter): C(n,r) = n! / (r! × (n-r)!) — ways to choose r items from n. These formulas are used in probability, statistics, and combinatorics.

Factorials in Calculus — Taylor Series

Factorials appear in Taylor series expansions used to approximate functions: e^x = 1 + x + x²/2! + x³/3! + ... sin(x) = x − x³/3! + x⁵/5! − ... cos(x) = 1 − x²/2! + x⁴/4! − ... These infinite series are how calculators and computers internally compute transcendental functions like e^x, sin, and cos.

Stirling's Approximation for Large Factorials

For very large n, computing n! directly requires enormous numbers. Stirling's approximation gives a close estimate: n! ≈ √(2πn) × (n/e)^n. This is used in statistical mechanics, information theory, and any field dealing with large combinatorial numbers. For programming, use logarithms of factorials (log(n!)) to avoid overflow when working with very large n.