A intersecção entre binômio de Newton e problemas de contagem

Abstract

This dissertation explored the intersection between Newton's Binomial, counting problems and generating functions, highlighting the importance of these concepts in combinatorial mathematics. Newton's Binomial, which allows the expansion of powers of binomials, is a fundamental tool for solving various combinatorial problems, providing a structure that facilitates the identification of numerical patterns and relationships. Initially, there is a theoretical review of Newton's Binomial, discussing its properties and applications in various contexts of mathematics. Then, there is an approach to how this concept directly relates to counting problems through practical examples that illustrate its effectiveness in solving problems involving combinations and permutations. In addition, generating functions are highlighted as a powerful tool for dealing with numerical sequences and series and how generating functions can be used in conjunction with Newton's Binomial to solve some combinatorial problems. The work also included the analysis of teaching activities that incorporate these concepts in high school, with the aim of promoting more dynamic and meaningful learning. It is noticeable that the integration of Newton's Binomial and generating functions into the school curriculum not only enriches students' mathematical knowledge, but also develops critical skills essential for problem-solving. This work aimed to contribute to the understanding of combinatorial mathematics and its application in educational contexts, encouraging future research in this area.