Factorial, Gamma Function & Stirling Calculator

Factorial Guide

Factorials and Counting

n! = n × (n−1) × (n−2) × ... × 2 × 1. 0! = 1 (by convention — one way to arrange nothing). Key values: 5! = 120. 10! = 3,628,800. 20! = 2.43×10¹⁸. 52! = 8.07×10⁶⁷ (more than atoms in the observable universe). Factorials grow faster than any polynomial or exponential function. Stirling's approximation (accurate for large n): n! ≈ √(2πn) × (n/e)ⁿ. For n=10: exact = 3,628,800. Stirling: √(62.83) × (10/2.718)¹⁰ = 7.926 × 3,598,696 ≈ 3,598,600. Error: 0.83%.

Double Factorial

Double factorial n!! = n × (n−2) × (n−4) × ... down to 1 or 2. For even n: n!! = n × (n−2) × ... × 4 × 2. For odd n: n!! = n × (n−2) × ... × 3 × 1. Examples: 6!! = 6 × 4 × 2 = 48. 7!! = 7 × 5 × 3 × 1 = 105. Note: n!! ≠ (n!)! — it does NOT mean the factorial of factorial n. Wallis product: relates double factorials to π: π/2 = (2·4·6·8·...)/( 1·3·5·7·...) = (2·2·4·4·6·6·...)/(1·3·3·5·5·7·...) Double factorials appear in probability distributions and integration of trigonometric functions.

Subfactorials and Derangements

A derangement is a permutation where NO element appears in its original position. !n = n! × Σ(k=0 to n) (-1)^k/k! ≈ n!/e. !0 = 1, !1 = 0, !2 = 1, !3 = 2, !4 = 9, !5 = 44. Classic problem: 'hat check' problem. n people each check in a hat. Hats are randomly returned. Probability no one gets their own hat: !n/n! → 1/e ≈ 0.3679. Remarkably, for n ≥ 5, the probability approaches 1/e regardless of n — approximately 36.79% of random shuffles are complete derangements.

Applications of Large Factorials

Card shuffling: 52! different orderings of a deck. If you could check one trillion orderings per second since the Big Bang (13.8 billion years), you would have checked approximately 4.36×10²⁹ orderings — a tiny fraction of 52! ≈ 8×10⁶⁷. Every properly shuffled deck has almost certainly never existed before. Password spaces: an 8-character password from 62 characters: 62⁸ ≈ 2×10¹⁴. An 8-character ALL DIFFERENT character password: P(62,8) = 62!/(62-8)! ≈ 1.36×10¹⁴. Genetics: the number of possible