rcombinations

In combinatorics, an r-combination of a finite set with n elements is a subset that contains exactly r elements, where the order of selection does not matter. The number of r-combinations is given by the binomial coefficient C(n, r) = n! / (r!(n − r)!). This value is defined for integers n ≥ 0 and 0 ≤ r ≤ n; if r is outside this range, the count is zero.

An example: from a set of four elements, the number of 3-element subsets is C(4, 3) = 4.

Key properties include symmetry and a recurrence relation. The symmetry property states C(n, r) = C(n, n − r), which reflects the idea that choosing r elements is equivalent to leaving out n − r elements. The recurrence relation, known as Pascal’s identity, is C(n, r) = C(n − 1, r) + C(n − 1, r − 1). Boundary values include C(n, 0) = C(n, n) = 1 for any n ≥ 0.

r-combinations connect to broader concepts in algebra and probability. They appear in the binomial theorem, where (1 + x)^n = sum from k = 0 to n of C(n, k) x^k. They also underpin counting arguments in probability and statistics, such as determining the number of possible subsets in a sample or the likelihood of selecting a given subset in a random draw.

There is a related concept of combinations with repetition (multiset combinations), counted by C(n + r − 1, r). This article focuses on combinations without repetition; for repetition allowed, the counting differs as described.