The study of permutations and combinations is at the root of several topics in mathematics such as number theory, algebra, geometry, probability, statistics, discrete mathematics, graph theory, and many other specialties.

Permutations

A permutation is an ordered arrangement of objects. For instance, the fraction is a permutation of two objects, whereas the combination to open a lock—23 L, 5 R, and 17 L—is a permutation of three objects. The ordered arrangements of objects likely dates all the way to the beginning of organizing and recording information.

Sometimes one is interested in knowing the number of permutations that are available from a collection of objects. Suppose a club has two candidates for president: Bob (B) and Janice (J); three candidates for secretary: Katy (K), Rob (R), and Harry (H); and three candidates for parliamentarian: Abe (A), Calvin (C), and Mary (M). In how many different ways...