rpermutations

R-permutations are a concept in combinatorics describing the number of ordered arrangements of r distinct elements drawn from a set of n distinct elements. In an r-permutation, the order of selection matters and no element can be repeated. The standard notation for the number of r-permutations of an n-element set is nPr or P(n, r). The counting formula is nPr = n! / (n − r)!, valid for integers n ≥ r ≥ 0. This arises from the idea that there are n choices for the first position, n−1 for the second, and so on, down to n−r+1 for the r-th position, giving the product n(n−1)…(n−r+1). When r = n, nPn = n!, the total number of full permutations of the set. When r = 0, nP0 = 1 by convention.

Example: From a group of five people, the number of ways to choose and arrange three is 5P3 = 5×4×3 = 60.

Relation to combinations: The number of r-combinations is nCr, which counts unordered selections. They relate by nPr = nCr × r!, since there are r! ways to order each chosen r-element subset. Conversely, nCr = nPr / r!.

Generalizations: If repetition is allowed, the number of r-length arrangements is n^r. If elements are not distinct or other constraints apply, counting requires different rules. R-permutations thus provide a foundational tool for counting ordered selections without repetition in finite sets.