faktoriale

In mathematics a factorial is an operation on a non‑negative integer n that results in the product of all positive integers less than or equal to n. It is denoted by an exclamation point, such as 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition 0! = 1, which keeps many combinatorial formulas valid.

The factorial grows very rapidly. For example 10! equals 3,628,800 and 20! is 2,432,902,008,176,640,000. Because of its growth rate, factorials often appear in approximations for natural logarithms and in Taylor series expansions, such as the exponential function e^x = Σ x^n / n!.

Factorials are fundamental in combinatorics. They give the number of permutations of n distinct objects, n! = P(n, n). They also allow construction of binomial coefficients via the relation C(n, k) = n! / (k! (n−k)!), which counts k‑element subsets of an n‑element set.

The operations of computing n! can be implemented recursively, iteratively, or using closed forms like Stirling’s approximation for large n. In computer science, factorials are used in algorithm analysis, particularly in counting the size of solution spaces for sorting, graph traversal, and backtracking algorithms.

Historically factorials were studied by early combinatorists such as Gerolamo Cardano and later by Isaac Newton and Leonhard Euler, who developed the gamma function to extend factorials to complex arguments via Γ(n+1) = n!.