Factorial Calculator

What a factorial counts

n! is the product of every integer from 1 to n, and it counts the number of ways to arrange n distinct items in order. Five books on a shelf can be arranged 5! = 120 ways: five choices for the first position, four for the second, and so on. That arrangement count is the definition's actual meaning, not a consequence of it.

Why 0! is 1

There is exactly one way to arrange nothing — the empty arrangement — so the count is 1. It also follows from the recursive definition, since n! = n × (n−1)! requires 1! = 1 × 0!, which forces 0! = 1. It is a necessary value, not a convention adopted for tidiness.

Growth outruns everything

Factorials grow faster than any exponential. 10! is 3.6 million, 20! is about 2.4 × 10¹⁸, and 70! exceeds 10¹⁰⁰ — more than the estimated number of atoms in the observable universe. This is why brute-force approaches to permutation problems become impossible so abruptly: 15 items is 1.3 trillion arrangements, and 20 is beyond any computer.

Where factorials appear

Permutations of n items taken r at a time are n!/(n−r)!. Combinations, where order does not matter, are n!/(r!(n−r)!) — the binomial coefficient behind Pascal's triangle and the binomial theorem. Factorials also sit in the denominators of the Taylor series for e^x, sine and cosine, which is why they show up in numerical analysis far from any counting problem.

Computing them safely

Standard integer types overflow quickly: 13! exceeds a 32-bit signed integer and 21! exceeds a 64-bit one. Some languages wrap silently to a negative number, giving a confidently wrong answer with no error. Anything past 20! requires arbitrary-precision arithmetic, and where only the magnitude matters, working with log factorials — summing logarithms rather than multiplying — avoids overflow entirely.

Non-integers and the gamma function

The factorial extends to non-integer and complex arguments through the gamma function, where Γ(n) = (n−1)! for positive integers. It gives meaning to expressions like the factorial of one half, which equals √π/2, and it is why statistical distributions with continuous parameters can be written in factorial-like form.