Factorial

Factorial, denoted by n!, is a function defined for non-negative integers n as the product of all positive integers up to n. That is, n! = 1 × 2 × ... × n, with 0! defined as 1. For example, 5! = 120. The factorial satisfies the recurrence n! = n × (n−1)!, with the base case 0! = 1.

In mathematics, factorial is used to count arrangements and selections of objects. It appears in formulas for binomial coefficients, such as n choose k = n!/(k!(n−k)!). Factorials also arise in probability, algebra, and in the series expansion of the exponential function, where e = sum from n=0 to ∞ of 1/n!.

Extension to non-integer values is provided by the gamma function, Γ(z), which satisfies Γ(n+1) = n! for non-negative integers n. The gamma function extends factorial to complex numbers (except at non-positive integers where it has poles).

Factorials grow very rapidly with n. For large n, Stirling’s approximation provides n! ~ sqrt(2πn) (n/e)^n, giving a practical estimate of their size. This rapid growth underpins many combinatorial and analytical calculations, including estimates of permutations and probabilities for large sets.

Related concepts include the double factorial n!!, and the rising and falling factorials, which generalize products of sequences of consecutive numbers. The factorial function is a fundamental tool in combinatorics, probability, and mathematical analysis.