Factorials

Factorials are a mathematical function defined for nonnegative integers, denoted by n!, and representing the product of all positive integers up to n. By convention, 0! is defined as 1. For example, 4! = 24 and 5! = 120.

The factorial function can be computed recursively: n! = n × (n − 1)!, with the base case 0! = 1. This makes it straightforward to derive values and is convenient in many proofs and algorithms.

Factorials have a clear combinatorial interpretation: they count the number of ways to arrange n distinct objects in a linear order, i.e., the number of permutations of n items. They also appear in the formula for binomial coefficients, nCk = n! / (k!(n − k)!), which count k-element subsets from a set of n elements. Factorials grow very rapidly with n.

Beyond integers, the factorial concept is extended by the Gamma function, Γ(z), which satisfies Γ(n+1) = n! for nonnegative integers. For large n, Stirling's approximation n! ≈ sqrt(2πn) (n/e)^n provides a useful estimate. Related products include the double factorial n!! and various multifactorials.

Factorials appear throughout mathematics, notably in series expansions such as e^x = sum_{n=0}^∞ x^n / n!, in probability, combinatorics, and asymptotic analysis.