Factorial Calculator (n!)

n! = 1 × 2 × … × n. Computes exactly up to 21! (number limit); approximate via Stirling for larger.

Inputs

Result

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General calculation reads

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How to use this calculator

  • Enter n (0 to 200 or so).
  • Read exact value or Stirling approximation.

About this calculator

n! ("n factorial") is the product of all positive integers up to n. 5! = 120; 10! = 3,628,800; 21! = 51,090,942,171,709,440,000 (last one fitting in JS Number). For larger n, this calculator uses Stirling's approximation: ln(n!) ≈ n ln n − n + ½ ln(2πn), accurate to <1% for n > 5. Factorials grow superexponentially: 100! has 158 digits, 1000! has 2568. Used in combinatorics (permutations), Taylor series, probability.

How it works — the formula

n! = n · (n − 1) · (n − 2) · … · 1 0! = 1 (empty product) Stirling: n! ≈ √(2πn) · (n / e)ⁿ

The factorial of a non-negative integer n is the product of all positive integers up to n. By convention 0! = 1 (the empty product). Factorials grow super-exponentially — n! beats every fixed exponential aⁿ for sufficiently large n. Stirling's approximation gives a tight asymptotic estimate that is accurate to within 1% by n = 10 and 0.01% by n = 100. Beyond integer arguments, the gamma function Γ(n) = (n − 1)! generalizes the factorial to real and complex numbers.

Sources: NIST DLMF §5 — Gamma function and factorials · NIST DLMF §5.11 — Asymptotic expansions / Stirling formula · OEIS A000142 — n!

Worked examples

Example 1
Small factorial
Inputs:
n = 5
Output:
5! = 5·4·3·2·1 = 120
Example 2
Combinatorics: card permutations
Inputs:
n = 52 (deck shuffles)
Output:
52! ≈ 8.066 × 10⁶⁷
Example 3
Stirling vs exact
Inputs:
n = 10
Output:
10! = 3,628,800; Stirling ≈ 3,598,696 (error ≈ 0.83%)

Limitations

  • Factorial is only defined here for non-negative integers; use the gamma function for non-integer or negative-real inputs.
  • IEEE 754 double precision overflows around 170! ≈ 7.3 × 10³⁰⁶; results above this are returned as Infinity.
  • For exact arbitrary-precision factorials, use a BigInt or symbolic-computation engine.
  • Stirling's approximation diverges as a series — taking too many terms makes accuracy worse.

Computed via the standard recursive product up to ~21!, then via lgamma+exp for larger inputs to avoid intermediate overflow.

Frequently asked

Why does it overflow at 21!?+
JavaScript Number is IEEE 754 double — 53-bit precision. 21! = 5.1 × 10^19, just past 2^53. Larger values lose precision.
Is 0! defined?+
Yes, 0! = 1 by convention (empty product). Important for the formula nCk to work for k=0.
Stirling accuracy?+
For n=10: 0.83% error. For n=100: 0.083%. For n=1000: 0.0083%. Each 10× n improves accuracy ~10×.
What is n! used for?+
Permutations: how many ways to arrange n items = n!. Combinations nCk = n!/(k!(n−k)!). Taylor series. Probability.
Beyond integers?+
Gamma function Γ(n) = (n−1)! generalizes to all real and complex numbers (except non-positive integers).

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