faktorieller

Faktoriell is a term used in mathematics and statistics to describe two related ideas: the factorial function and the factorial design of experiments. The plural form faktorieller is used when referring to multiple factorials or multiple faktoriell designs.

In mathematics, a factorial, written n!, is the product of all positive integers from 1 to n. The definition extends to 0! = 1. Factorials grow rapidly and are used to count permutations: the number of ways to arrange n distinct items is n!. The factorial function can be extended beyond integers through the gamma function, with Γ(n+1) = n!. The double factorial n!! and other multi-factorials are variations used in specific combinatorial formulas.

In combinatorics, factorials underpin binomial coefficients, since n choose k equals n!/(k!(n−k)!). They also appear in probability distributions and series expansions, such as the coefficients in the expansion of (1+x)^n and in Poisson and normal approximations for large n.

In statistics and experimental design, a faktoriell design (factorial design) analyzes the effects of several factors by varying them simultaneously. A full faktoriell design with k factors at two levels has 2^k experimental runs; each factor's main effect and interactions with other factors can be estimated. Fractional faktoriell designs reduce the number of runs but may alias effects, which reduces interpretability. The choice of design depends on goals, levels, and resource constraints.