Let’s learn to count

In this chapter, after a quick review of the basic concepts of set theory, we introduce the fundamental notions and principles of combinatorics. Even though its contents are elementary, we warmly suggest to take a look at the chapter. Our approach consists in trying to describe every combinatorial problem by means of sets of (ordered) sequences, or (unordered) collections, and their dual concepts of sharings and compositions. Computations are successively done via some basic fundamental tools like the Multiplication and the Division Principle. A rigorous and effective formulation of these principles, in particular of the Multiplication Principle, is of fundamental importance for their correct application. Indeed they constitute, at the same time, the royal way to solve combinatorial problems, and the main source of errors, when misused. We conclude the section with a brief discussion of uniform probability on finite sample spaces, which is here just a way to express combinatorial results in probabilistic terms.