Last updated: 2026-05-22

Definition

The factorial of a non-negative integer $n$, written $n!$, is the product of every positive integer from 1 up to $n$:

Common values:

The value of 0!

The definition $0! = 1$ looks surprising at first, but it follows naturally from the recurrence $n! = n \times (n-1)!$. Setting $n = 1$ gives $1! = 1 \times 0!$, which requires $0! = 1$. Conceptually, there is exactly one way to arrange zero objects: do nothing. This convention also keeps the formulas for permutations and combinations correct at their boundaries.

Growth rate

Factorials grow faster than any exponential function. By $n = 13$ the result already exceeds 6 billion ($13! = 6{,}227{,}020{,}800$). By $n = 20$ the result has 19 digits. This is why the calculator caps at 20 — beyond that, exact integer results would overflow standard 64-bit integers.

Stirling's approximation gives a useful estimate for large $n$:

Applications

Counting arrangements. The number of ways to arrange $n$ distinct objects in a row is exactly $n!$. For example, five books can be shelved in $5! = 120$ different orders.

Permutations and combinations. Both formulas are built from factorials:

Probability. Factorials appear in the binomial distribution, the Poisson distribution, and anywhere you compute discrete probabilities over ordered outcomes.

Calculus. Taylor series use factorials in the denominator to control the growth of successive terms. The exponential series is $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$.

Practical notes

Integer output only. For $n \leq 20$, factorials are exact integers. This calculator returns the precise value with no rounding.

Permutations and combinations. The Permutation Calculator — P(n, r) computes $P(n, r) = n! / (n-r)!$, and the Combination Calculator — C(n, r) computes $C(n, r) = n! / (r!(n-r)!)$ directly, without computing the factorials separately.