Enumerative combinatorics is an area of
combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting
combinations and counting
permutations. More generally, given an infinite collection of finite sets
Si indexed by the
natural numbers, enumerative combinatorics seeks to describe a
counting function which counts the number of objects in
Sn for each
n. Although counting the number of elements in a set is a rather broad
mathematical problem, many of the problems that arise in applications have a relatively simple
combinatorial description. The
twelvefold way provides a unified framework for counting
permutations,
combinations and
partitions.