Kombination

Kombination, in German mathematical usage, refers to the concept of a set selection: a combination is a selection of k elements from a finite set of n distinct elements in which the order of selection does not matter. The number of such selections is given by the binomial coefficient C(n, k), often described in English as “n choose k” and written as C(n, k) = n! / (k!(n - k)!).

When repetitions are not allowed, this count applies directly. For example, choosing 3 items from a set of 5 yields C(5, 3) = 10 combinations.

Combinations with repetition, also known as multisets, allow the same element to be chosen more than once. The number of size-k combinations from n types is C(n + k - 1, k).

Applications of combinations appear in probability, statistics, card games, lottery calculations, and the design of experiments. They are closely related to the binomial theorem, which expresses (x + y)^n as a sum involving binomial coefficients, and to more advanced topics such as hypergeometric distributions and combinatorial designs.

Historically, the study of combinations contributed to the development of probability and enumerative methods. The term Kombination is standard in German mathematics and is used interchangeably with the English term combination in translated or bilingual texts.