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Methods
Abstract
The chapter explores the fundamental concepts of combinatorics, presenting essential methods and models for the analysis of combinatorial structures. The chapter discusses classical combinatorics, focusing on permutations, combinations, and partitions. It introduces the perspective of J.G. Dubois, who emphasizes combinatorial structures rather than mere enumeration, classifying problems into selections, distributions, and partitions. The fundamental principle of product and sum, the principle of equality and inclusion-exclusion, which allow us to manage counting problems and set structures, are examined in detail. Additional combinatorial tools discussed include generating functions, which transform combinatorial problems into algebraic problems, and binomial coefficients, which are fundamental in the calculus of combinations and related to Pascal's triangle. The chapter also highlights combinatorial models such as shafts and the pigeonhole principle. Finally, the computational aspects of combinatorics are explored through algorithms.
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