Kombinatsiooonide

Kombinatsiooonide refers to the concept of selecting subsets from a finite set where the order of selection does not matter. In combinatorics, these selections are called combinations. The standard Estonian term is kombinatsioonide, but the form kombinatsiooonide may appear in stylized spelling or typographical variants.

The central object is the binomial coefficient, written as C(n, k) or "n choose k." It counts the number of k-element subsets of an n-element set and is given by C(n, k) = n! / (k!(n − k)!). For example, choosing 3 from 5 yields C(5, 3) = 10. If k > n, the value is zero; if k = 0 or k = n, the value is 1. Combinations are distinguished from permutations, where order matters; the number of permutations of k items from n is P(n, k) = n! / (n − k)!.

Combinations with repetition allow selecting k items from n types with unlimited repeats. The number is C(n + k − 1, k), a result often derived via the stars-and-bars method. This extends to multisets and to counting distinct selections when repetition is allowed.

Connections to broader theory include the binomial theorem, which expresses (a + b)^n as a sum of terms involving C(n, k). Pascal’s triangle provides a visual and recursive means to compute binomial coefficients. Applications of kombinatsiooonide cover probability (calculating the likelihood of specific hands or outcomes), statistics, lottery designs, card games, genetics, and algorithmic generation of combinations.

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