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In how many ways…? Enumerative combinatorics is the art of counting different configurations in problems of a discrete nature.

This book offers an elementary introduction to the fascinating and challenging field of combinatorics. In the first part, it presents the fundamental results that, in the author’s opinion, every student of mathematics or mathematics education should be familiar with and study.

The theory is presented in a summarized and condensed form, with a particular focus on proofs of an intrinsically combinatorial nature. The author aims to avoid non-combinatorial methods, such as mathematical induction, with minimal exceptions. Instead, the work favors alternative proofs that rely on purely combinatorial arguments. This approach highlights the wealth of resources available for reasoning within this field of mathematics, where proofs can be both ingenious and informal. For example, well-known results like Newton's binomial and multinomial theorems are demonstrated concisely using only elementary combinatorial techniques.

It is important to recognize that enumerative combinatorics is a vast and intricate field, with a significant body of deep and ongoing research. This book is designed to provide an introductory overview of the general theory, focusing on presenting the most important and intriguing results to the student.

For readers interested in exploring combinatorics in greater depth, I recommend Claude Berge's esteemed work Principles of Combinatorics. This classic text is renowned for its clarity and accessibility, making it an excellent resource for further study in the field.

One of the objectives of this work is to address the lack of accessible teaching material on the fundamentals of combinatorial theory. While there are highly specialized and well-regarded treatises in the field of combinatorics, they are often too advanced and inaccessible for students in the early years of mathematics and mathematics education. This book is specifically designed for these students, aiming to provide an accessible and comprehensive introduction to the subject.

The book is organized into two parts. The first part comprises nine chapters: the first eight chapters cover fundamental topics in enumerative combinatorics and include a variety of exercises to challenge and engage the reader. The ninth chapter provides solutions to all the odd-numbered exercises from the preceding eight chapters.

The second part of the book delves into more specialized topics. Chapters 10 and 11 offer in-depth studies of Hadamard matrices and combinatorial designs, respectively, presenting both the foundational theory and some unresolved theoretical problems. Lastly, Chapter 12 introduces transfinite enumerative combinatorics, using the intriguing problem of Hercules versus the Hydra as a gateway to this advanced area of study.

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