Factorial Calculator

Factorial (n!)

The factorial of a non-negative integer n is the product of all positive integers up to n: n! = 1·2·3·…·n, with the convention 0! = 1 (justified by the gamma function). It grows very fast: 10! = 3,628,800; 13! already exceeds 6 billion (Int32 overflow); 20! ≈ 2.4×10¹⁸; 100! ≈ 9.33×10¹⁵⁷. For large n, Stirling's approximation is handy: n! ≈ √(2πn) · (n/e)^n. The gamma function extends factorial to real and complex numbers: Γ(n+1) = n!, which lets you compute things like (1/2)! = √π/2.

Applications and context

Factorials appear in combinatorics (C(n,k) = n! / (k!(n−k)!), A(n,k) = n! / (n−k)!), Taylor series (e^x = Σ x^n/n!), probability (poker hands, lottery odds like Mega-Sena), and algorithm complexity (brute-force Travelling Salesman is O(n!)). They're a staple of high-school combinatorial analysis (ENEM-style problems).

FAQ

Why is 0! equal to 1? By convention, so that formulas like C(n,0) = n!/(0!·n!) = 1 work. Gamma also gives Γ(1) = 1.

Is there a factorial of a negative number? Not for integers — the gamma function has poles at 0, −1, −2, …. But for non-integer negatives like −0.5, Γ is defined and finite.

How big can JavaScript handle exactly? Up to 18! using Number; beyond that you lose precision. Use BigInt for exact results past ~21!.