The main goal of the two authors is to help undergraduate students
understand the concepts and ideas of combinatorics, an important realm
of mathematics, and to enable them to ultimately achieve excellence in
this field. This goal is accomplished by familiariz- ing students with
typical examples illustrating central mathematical facts, and by
challenging students with a number of carefully selected problems. It is
essential that the student works through the exercises in order to build
a bridge between ordinary high school permutation and combination
exercises and more sophisticated, intricate, and abstract concepts and
problems in undergraduate combinatorics. The extensive discussions of
the solutions are a key part of the learning process. The concepts are
not stacked at the beginning of each section in a blue box, as in many
undergraduate textbooks. Instead, the key mathematical ideas are
carefully worked into organized, challenging, and instructive examples.
The authors are proud of their strength, their collection of beautiful
problems, which they have accumulated through years of work preparing
students for the International Math- ematics Olympiads and other
competitions. A good foundation in combinatorics is provided in the
first six chapters of this book. While most of the problems in the first
six chapters are real counting problems, it is in chapters seven and
eight where readers are introduced to essay-type proofs. This is the
place to develop significant problem-solving experience, and to learn
when and how to use available skills to complete the proofs.
