fakultetsberegning

Fakultetsberegning, or factorial calculation, concerns the factorial function defined for non-negative integers n by n! = 1 × 2 × ... × n, with 0! = 1. The value grows rapidly as n increases, and factorials are widely used in counting and algebraic problems.

Historically, the notation n! was introduced by French mathematician Christian Kramp in 1808. The concept appears earlier in work on permutations and combinatorics, but Kramp established the standard symbol used today.

Factorials are central in combinatorics: the number of ways to arrange n distinct objects is n!, and the binomial coefficient n choose k, used for counting subsets, is n!/(k!(n−k)!). Factorials also appear in series such as the expansion of e, with e = sum_{k=0}^∞ 1/k!, and in various probability distributions and analytic formulas.

Extensions include the gamma function, which generalizes factorials to non-integer values: Γ(n+1) = n! for integers, with Γ(z) defined for complex numbers with positive real part. This provides a continuous extension of the discrete factorial function.

Computation and approximation: simple factorials are computed by multiplying 1 through n; for large n, Stirling's approximation n! ~ sqrt(2πn)(n/e)^n provides a good estimate. In computing exact factorials, big-integer arithmetic is used to avoid overflow, and logarithms or the log-gamma function can help manage very large values.

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