Permutations and Combinations

Patrick McAtee, Tom Rice, and Jesse Whidden

## Introduction

Do you know anyone that doesn’t know how to count? Most people learn at an early age how to count a set of objects one by one. However, there are problems where counting in the typical fashion will not suffice. These are problems that deal with looking at the number of ways to combine objects. For small numbers it is often possible to count possibilities one by one, but a small increase in choices soon yields a large increase in outcomes. For these situations, a branch of math called combinatorics has been developed.

Combinatorics arose in the seventeenth century in response to the rise in popularity of card games. People wanted to know better ways to count cards and determine probabilities because this would give them an advantage. This first application of combinatorics might have been quite lucrative for some early card players, but wasn’t productive.

There are many practical applications of advanced counting techniques. Some specific ones relating to Permutations and Combinations are figuring out if there are enough telephone numbers to serve a given population, or likewise for license plate numbers. If one needs to know the number of ways of combining things, combinatorics can be used to figure out how many possible arrangements there are.

This tutorial gives a basic introduction to Permutations and Combinations, a branch of combinatorics, and also delves into some of the deeper topics and applications of these methods. It will also explain simple ways to generate permutations and combinations, and also deal with what to do when repetition needs to be considered.

(This tutorial is based on Discrete Mathematics and its Applications, by Kenneth H. Rosen.)