Permutations

Permutations refer to the different possible arrangements of a set of objects in a sequence where the order of the objects matters. Two arrangements that differ only by order are considered distinct.

If the set has n distinct objects and all are used, the number of permutations is n!, the factorial of n. For example, the letters A, B, C can be arranged in six ways: ABC, ACB, BAC, BCA, CAB, CBA.

If k objects are chosen from n distinct objects without repetition and the order matters, the number is P(n, k) = n!/(n−k)!. Example: from five students, the first three positions can be filled in 5×4×3 = 60 ways.

If repetition is allowed and k objects are chosen from n distinct types, the number of ordered arrangements is n^k. Example: a three-character code from ten digits has 10^3 = 1000 possibilities.

When the items include repeats (a multiset), the number of distinct permutations of all n items is n! divided by the factorials of the multiplicities: n!/(n1! n2! ...). Example: the letters A A B yield 3!/(2!1!) = 3 permutations: AAB, ABA, BAA.

These counts rely on the factorial function, with 0! defined as 1. Permutations have applications in seating people, scheduling, ranking, and coding, where the order of selection or arrangement is important.