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Factorial Calculator

Compute n! with full step expansion — exact even for huge numbers.

5! = 120

5! = 5 × 4 × 3 × 2 × 1 = 120

5! has 3 digits.

Definition: n! = n × (n−1) × … × 2 × 1, with 0! = 1 by definition.

Uses: counting permutations and combinations in probability and statistics.

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Where Factorials Come From

Ask “in how many orders can 5 books sit on a shelf?” — the first slot has 5 choices, the next 4, then 3, 2, 1. That is 5! = 120. Factorials count permutations, and they explode fast: 10! is already 3.6 million, and 20! exceeds 2 quintillion.

The Two Facts That Confuse Everyone

  • 0! = 1 by definition — an empty product equals 1, which keeps combination formulas working.
  • Factorials only apply to whole numbers. For decimals, mathematicians extend the idea with the Gamma function, where n! = Γ(n+1).

Factorials in Probability

Combinations (nCr) and permutations (nPr) are built from factorials — lottery odds, poker hands, and committee selections all reduce to factorial division. Try the probability calculator to put them to work.

Frequently Asked Questions

What is a factorial?

The factorial of n (written n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

What is 0 factorial (0!)?

0! = 1 by definition. This keeps formulas like nCr consistent — there is exactly one way to arrange zero objects.

How do you calculate large factorials like 100!?

Large factorials have far too many digits for normal calculators. This tool uses exact BigInt arithmetic to compute factorials up to 1000! precisely.

Is this factorial calculator free?

Yes — every tool on Math Solver Free is free with no sign-up, and all calculations run in your browser.