femfaktoriale

Femfaktoriale refers to the factorial of five, commonly written as 5!. In mathematical terms, femfaktoriale is the product of all positive integers up to five: 5! = 5 × 4 × 3 × 2 × 1 = 120. The term is often used in Norwegian discussions of factorials and combinatorics.

In general, the factorial function n! is defined for non-negative integers n by n! = n × (n−1)!, with the base case 0! = 1. This yields the sequence 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, and so on. Factorials grow rapidly and satisfy the recurrence n! = n × (n−1)!.

Factorials have wide applications in counting and probability. They count the number of permutations of n distinct objects (n! possible orders). They appear in the calculation of binomial coefficients, in series expansions such as the exponential series e^x = sum from n=0 to ∞ of x^n/n!, and in various counting problems and combinatorial formulas.

Extensions and limitations: the standard factorial is defined only for non-negative integers. Negative integers do not have factorials in the usual sense. The gamma function provides a continuous extension of the factorial, with Γ(n+1) = n! for non-negative integers, and Γ(z) defined for most non-integer values as an extension of the factorial concept.

Historically, the factorial notation and concept were popularized in the early 19th century, with the exclamation mark attributed to mathematician Christian Kramp (around 1808) to denote the product of consecutive integers.