Permutation Calculator

A permutation is an arrangement in which order matters. If three finalists can receive gold, silver, and bronze medals, choosing the same three people in a different medal order creates a different outcome. This distinguishes permutations from combinations, where only membership matters. The permutation calculator counts ordered selections from a total of n available items, taking r positions at a time, and supports both the standard no-repetition model and a repetition-allowed model. Without repetition, an item can be used only once. There are n choices for the first position, n minus one choices for the second, and so on until r positions have been filled. Multiplying those counts gives P(n,r) = n! / (n-r)!. For five items taken three at a time, the result is 5 × 4 × 3 = 60. The calculator evaluates this falling product directly, avoiding unnecessary factorial operations while producing the same exact integer for practical inputs. With repetition, every position can use any of the n available items, including an item already used in an earlier position. The multiplication rule then gives n choices for each of r positions, so the count is n raised to the power r. A three-character code formed from four allowed symbols has 4 × 4 × 4 = 64 possible ordered codes. Repetition is therefore appropriate for PINs, strings, repeated trials, replacement sampling, and other situations where each choice restores the full set of options. Permutations appear throughout probability, computing, operations research, scheduling, and everyday planning. They count race finishing orders, seating plans, task sequences, routes, ranked ballots, password candidates, and assignments to distinct roles. Correctly identifying whether order matters is the essential first step. Selecting Alice and Ben for an unordered committee is one combination, but assigning Alice as chair and Ben as secretary differs from assigning Ben as chair and Alice as secretary, so those role assignments are permutations. The values n and r must be non-negative whole numbers because they represent counts. Under the no-repetition rule, r cannot exceed n because there are not enough distinct items to fill the positions. The case r = 0 has one result: the empty arrangement. Large permutation counts grow extremely quickly, so browser number formatting may eventually use an approximate floating-point value outside the safe integer range. For ordinary classroom, probability, and planning questions, the displayed result offers a fast and transparent count. Before using a result as a probability denominator, make sure all arrangements are equally likely and that the chosen repetition rule matches the experiment. Restrictions such as forbidden positions, required symbols, indistinguishable objects, or repeated objects in the original collection require additional methods. For the standard nPr scenarios described here, this calculator provides the canonical formula and an immediate answer.