Exploring Combinatorial Mathematics

by Richard Grassl & Oscar Levin

Exploring Combinatorial Mathematics is a free and open source textbook suitable for an upper level undergraduate or Master's level discrete math or combinatorics course, especially if the intended audience is pre-service or in-service secondary teachers. The text borrows material from Bogart's Combinatorics through Guided Discovery and Levin's Discrete Mathematics: an Open Introduction, falling somewhere between these in terms of difficulty.

Get the book

The entire book is available for free as an interactive online ebook. This should work well on all screen sizes, including smart phones. Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links. Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer.

For offline use, a free pdf version, suitable for reading on a tablet or computer, is available for download. This should be searchable and easy to navigate using embedded links. Hints and solutions (when available) can be accessed by clicking on the exercise number, and clicking on the number of the hint or solution will bring you back to the exercise.

If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. This should be cheaper than printing the entire book and binding it yourself. Page numbers match the pdf version.

About the book

This book was written specifically to be used as the textbook for the Master's level Discrete Mathematics course at the University of Northern Colorado. This course is part of a MA in Mathematics with a Teaching Emphasis. Most of the students in the course are current secondary math teachers. This intended audience has influenced the style and content of the book in a few important ways.

First, we acknowledge that not everyone reading this book will be immediately familiar with the content of a standard undergraduate discrete mathematics course. Little is assumed about the reader's previous work in the subject, beyond a general understanding of how abstract mathematics proceeds, as well as some basic ability with mathematical proof. For the reader completely unfamiliar with these and the basic objects of mathematical study (sets and functions), background material is included in an Appendix.

Topics have been selected to illustrate larger concepts of interest to secondary teachers. We have put an emphasis on understanding simple concepts deeply and in more than one way. Although some topics intersect secondary curriculum, most of the questions here are at a higher level. Still, the problem solving strategies and big ideas illustrated by our questions have applications to secondary mathematics. This emphasis is quite different than other mid level discrete and combinatorics textbooks, since we are not preparing our readers to begin a career in research mathematics.

While this course is not a course on teaching mathematics, we have tried to model good pedagogical practice. As you will see, almost all of the textbook consists of Activities and Exercises that guide students to discover mathematics for themselves. This will require quite a bit more work both from students and instructors, but we strongly believe that the best way to learn mathematics is by doing mathematics. Most of all, discovering mathematics is fun.


Exploring Combinatorial Mathematics is licensed under the GFDL (GNU Free Documentation License).

Permission is granted to copy, distribute and/or modify the document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in Appendix E of the text.