faktoriaal

Faktoriaal is the factorial function in mathematics, known as faktoriaal in Estonian. For a non-negative integer n, n! is defined as the product 1 × 2 × ... × n, with 0! = 1 by convention.

An important extension defines factorial for real or complex numbers via the gamma function: x! = Γ(x+1). This extends the notion to non-integer values, while Γ has poles at non-positive integers, so a direct factorial is not defined there.

Key properties include the recurrence n! = n × (n−1)!, with 0! = 1. The function grows very rapidly and is fundamental in counting and combinatorics. Combinatorially, n! counts the number of permutations of n distinct objects. It also underpins binomial coefficients: the number of k-element combinations from n elements is given by nCk = n!/(k!(n−k)!). In probability and series, factorials appear in expansions such as the Taylor series of e^x and in many combinatorial identities.

Approximation and asymptotics are often used for large n: Stirling’s formula n! ~ sqrt(2πn) (n/e)^n provides a practical estimate. Special values include 0! = 1 and 1! = 1.

Limitations include that the standard factorial is defined only for non-negative integers; the gamma function provides a broad generalization, but not all properties of integer factorials carry over verbatim.