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Combinatorics: A Problem Oriented Approach

Daniel A. Marcus
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Ruth Michler
, on

Combinatorics: a problem oriented approach is a book on Combinatorics that mainly focuses on counting problems and generating functions. By restricting himself to an accomplishable goal, without attempting to be encyclopedic, the author has created a well-focused, digestible treatise on the subject.

According to the author's preface, the book is based on lecture notes on a course on Combinatorics taught by the author at California Polytechnic, Pomona, for more than twelve years. The intended audience is Computer Science and Mathematics majors in their junior or senior year or interested scientists, mathematicians, and amateur mathematicians who might want to use this book for self-study. Strictly speaking, there are no prerequisites for reading this book. However, mathematically less sophisticated readers might struggle with some of the terminology and notation. For example, factorials, binomial coefficients, summation notation are all used without definition. The students at Cal Poly Pomona might not experience such problems, but a more general audience might.

Minor complaints

On the reviewer's wish-list would have been a table of necessary definitions and binomial identities. A more formal definition of probability might also be appropriate. In the first problem the reader is asked what is the probability that a five letter word with letters selected from the set {A,B,C} does not contain the letter A, but no definition of probability is given. Such omissions can easily be patched up in class, when there are questions, but they are flaws in a book which intends to be accessible to students and suitable for self-study.

This book is very strong on heuristics (see below), but does not offer any of the proofs or formal definitions that a more rigid course on combinatorics usually includes. At the reviewer's institution the equivalent course is used to introduce proof techniques (especially induction). Thus the book under review would not be appropriate as a textbook for our course.

Strong points

The book is unique in that it moves from questions to the theory. It begins by asking a motivating question that leads one to introduce a basic concept, such as counting with repetition and with order, in the process of finding an answer. Thus, the book is strong on motivation and has a wealth of problems of varying degrees of difficulty. This means that even instructors who would not want to choose this book as a textbook, because of its lack of proofs, might want to choose this book as a problem source. In fact, with adequate coaching to help with some of the notation this book could also be used as a Putnam preparation or for Math summer camp instruction.

Motivated students (especially those on a self-study course) will appreciate the solutions to some problems and the guide as to their interdependence. The choice of topics is excellent. This book is one of the few that treats the Pólya-Redfield counting method and recurrence relations, although the treatment of recurrence relations is somewhat incomplete.


This book is certainly worth the $20 price tag

Ruth Michler is assistant professor of mathematics at the University of North Texas.