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 "s" indexed by the natural numbers, enumerative combinatorics seeks to describe a "counting function" which counts the number of objects in "s" 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.