Why would anyone want to solve combinatorics problems? The best reason has to be because it's simply fun. It's also a great way to sharpen your problem solving skills and give your brain a good workout. An hour spent solving combinatorics problems is better than an hour spent playing chess since it's not only fun but it gives you skills you can use in mathematics, computer science, physics, biology, etc.

Let's define what we mean by combinatorics. Much of combinatorics is based on the idea of counting. This may involve simply counting the number of elements in a set. It sounds simple but it's not a matter of just looking and counting. The set is usually only defined as elements that meet some condition or have some property. You could, in principle, construct the set from the definition and then count the number of elements it contains. When the number of elements runs into the thousands, millions, billions, or becomes infinite then things become quite tedious. So to solve combinatorics problems you often need some insight, ingenuity, and the ability to turn the problem into a different form that is more easily solved. What could be more fun than that?

This is not meant to be a textbook on combinatorics but there is enough introductory material so that even someone with little or no prior exposure to the subject can get something out of it. Some familiarity with the concept of sets, subsets, factorials, and basic algebra is all that is required. We start with some definitions in the introduction along with a guide to solving counting problems. The guide is a list of some of the most common counting problems. There is an equation for each problem and a set of equivalent descriptions of the problem. This is followed by short sections that explain some of the most basic principles in combinatorics. They were chosen because they are used in the problems but they are by no means exhaustive.

After the introduction comes the problems and exercises. The problems are generally easier and shorter than the exercises and increase in complexity as you go. Some of the exercises can be quite involved and may take some time to fully work out. A computer algebra system capable of dealing with very large numbers may be helpful for some of the problems and exercises. Each problem and exercise is fully worked out in detail. Most of the problems and exercises are modernized versions of the problems and exercises found in the book: Choice and Chance by W. A. Whitworth (see Further Reading at the end of the book). Happy problem solving.

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Stefan Hollos and J. Richard Hollos
Exstrom Laboratories LLC
Longmont, Colorado, U.S.A.
January 2013