In this paper, we will review two fundamental counting methods: inclusionexclusion and pigeonhole principles. The inclusion-exclusion principle considers the elements of the sets satisfied some conditions, and avoids repeat counting by disjoint sets. We also use the inclusion-exclusion principle to solve the problems of Euler phi function and the number of onto functions in number theory, and derangement and the number of nonnegative integer solutions of equations in combinatorics. We derive the closed-form formula to those problems. For the forbidden positions problems, we use the rook polynomials to simplify the counting process. We also show the form of the inclusion-exclusion principle in probability, and use it to solve some probability problems. The pigeonhole principle is an easy concept. We can establish some sets and use the pigeonhole principle to discuss the extreme value about the number of elements. Choose the pigeons and pigeonholes, properly, and solve problems by the concept of the pigeonhole principle. We also introduce the Ramsey theorem which is an important application of the pigeonhole principle. This theorem provides a method to solve problems by complete graph. Finally, we give some contest problems about the inclusion-exclusion and pigeonhole principles to show how those principles are used.

Many countries are reporting of difficulties in exact education at schools: mathematics, informatics, physics etc. Various methods are proposed to awaken and preserve students’ interest in these disciplines. Among them, the simplification, accent on applications, avoiding of argumentation (especially in mathematics) etc. must be mentioned. As one of reasons for these approaches the growing amount of knowledge/skills to be acquired at school is often mentioned. In this paper we consider one of the possibilities to integrate partially teaching of important chapters of discrete mathematics and informatics not reducing the high educational standards. The approach is based on the identification and mastering general combinatorial principles underlying many topics in both disciplines. A special attention in the paper is given to the so-called “pigeonhole principle” and its generalizations. In folklore, this principle is usually formulated in the following way: “if there are n + 1 rabbits in n cages, you can find a cage with at least two rabbits in it“. Examples of appearances of this principle both in mathematics and in computer science are considered.

Many countries are reporting of difficulties in exact education at schools: mathematics, informatics, physics etc. Various methods are proposed to awaken and preserve students’ interest in these disciplines. Among them, the simplification, accent on applications, avoiding of argumentation (especially in mathematics) etc. must be mentioned. As one of reasons for these approaches the growing amount of knowledge/skills to be acquired at school is often mentioned. In this paper we consider one of the possibilities to integrate partially teaching of important chapters of discrete mathematics and informatics not reducing the high educational standards. The approach is based on the identification and mastering general combinatorial principles underlying many topics in both disciplines. A special attention in the paper is given to the so-called “pigeonhole principle” and its generalizations. In folklore, this principle is usually formulated in the following way: “if there are n + 1 rabbits in n cages, you can find a cage with at least two rabbits in it“. Examples of appearances of this principle both in mathematics and in computer science are considered.