Kombinations

Kombinations refers to the mathematical concept of combinations, and in everyday German usage also to any selection of items where the order does not matter. In combinatorics, a k-element combination from a set of n distinct elements is a k-element subset. The number of such combinations is given by the binomial coefficient C(n, k) = n! / (k!(n−k)!). This formula applies when 0 ≤ k ≤ n; outside this range the count is zero. When repetitions are allowed, i.e., combinations with repetition, the number of multisets is C(n+k−1, k).

Example: from a set of five elements, choosing three yields C(5,3) = 10. In probability and statistics, combinations describe outcomes where the order is irrelevant, such as selecting a committee or drawing cards without regard to order. In algebra, the binomial theorem uses these coefficients: (x + y)^n = Σ C(n, k) x^{n−k} y^k.

These coefficients appear in Pascal’s triangle and satisfy identities such as C(n, k) = C(n−1, k) + C(n−1, k−1). Combinations are contrasted with permutations, where the order of selection matters and the count is P(n, k) = n!/(n−k)!.

In general usage, "Kombination" can refer to any pairing or configuration of features, colors, or components. See also permutations; combinations with repetition; factorial; binomial theorem; Pascal’s triangle; combinatorics.