Princípios básicos e avançados sobre análise combinatória e uma proposta didática com o software SageMath

Abstract

Combinatorial Analysis is frequently regarded by both students and teachers as a chal-lenging subject. This difficulty often stems from the use of inadequate methods and approaches that, rather than fostering logical reasoning, rely almost exclusively on the mechanical application of formulas. In this study, we present fundamental counting princi-ples such as the Bijective, Additive, and Multiplicative Principles, as well as Permutations and Combinations (both simple and with repetition). We also explore advanced count-ing techniques, including the Inclusion-Exclusion Principle, Derangements, Kaplansky’s Lemmas, Recurrence-Based Counting, and the Pigeonhole Principle. Furthermore, we provide a section on counting strategies and another addressing the most common erros encountered in solving combinatorial problems. As an application of these concepts, we include solved problems from the Brazilian Mathematics Olympiad for Public and Private Schools (OBMEP) and the National High School Exam (ENEM). Finally, we propose the integration of a technological tool to support the teaching of Combinatorial Analysis in secondary education. To this end, we developed a didactic sequence of activities using the open-source software SageMath, which displays all possible grouping configurations and calculates their quantities. The activities were designed with a progressive level of difficulty, enabling students to gradually build the reasoning required to understand the underlying concepts and thereby achieve consistent learning outcomes.