The factorial of a non-negative integer \(n\), written \(n!\), is the product of all positive integers up to \(n\): \(n! = 1 \times 2 \times 3 \times \cdots \times n\). By convention \(0! = 1\).
Factorials count the number of orderings (permutations) of \(n\) distinct items. They appear at the heart of combinatorics: binomial coefficients \(\binom{n}{k} = n! / (k! (n-k)!)\), the exponential function's Taylor series, and the Gamma function which extends factorials to non-integer arguments.
Factorials grow extremely fast: \(20! \approx 2.4 \times 10^{18}\), beyond a 64-bit integer. Stirling's approximation \(n! \approx \sqrt{2 \pi n}(n/e)^n\) gives a tractable closed-form estimate.