Combinatorics

Combinatorics is the study of counting.

Introductory combinatorics

The two most basic and fundamental ideas are that of permutations and combinations. In essentials, the permutation is the number of ways to create a subset of a larger set if order matters (i.e. A, B, C is different from A, C, B). Similarly, the combination is the number of ways to create a subset of a larger set if order does NOT matter (i.e. A, B, C is the same as A, C, B).

Factorials

A permutation of a set of r objects is any rearrangement of the r objects with regard to order. To find how many ways we can do this, note that for the first of the r elements, we have n different objects we can choose from. For the second element there are (n-1) objects we can choose, (n-2) for the third and so on. In general, the number of ways to permute r objects from a set of n is given by (by the Fundamental Counting Principle

The number of combinations of r objects from a set of n total is the number of ways the r objects can be arranged with regard to order.

 Consider the set of letters A, B, and C.  There are P(3,3)=6 different permutations of those letters.  Since order doesn't matter with combinations, there is only one combination of those three.  In general, since for every permutation of r objects from n elements P(n,r), there are  more ways to permute them than to choose them.  We have r!C(n,r)=P(n,r) C(n,r) or C(n,r)

Intermediate combinatorics

An important result of counting techniques is the formulation of the Principle of Inclusion-Exclusion (PIE).