The problem of derangements and the inclusion-exclusion formula: from Montmort to the present day

This paper examines the development of the inclusion-exclusion principle and the method for solving the derangement problem from a historical perspective. Pierre Montmort was the first to formulate the derangement problem in a specific case. Not only did he derive a recurrence relation that later became a classic for counting the number of derangements, but he also thoroughly studied their properties. His work became a significant step in the development of combinatorics and probability theory. Later, Leonard Euler provided a more general and elegant solution to the problem using the inclusion-exclusion principle, which has since become standard in modern combinatorics textbooks. Thus, the explicit formula for the number of derangements was indeed first presented in Montmort's works. Other renowned mathematicians, such as Augustin-Louis Cauchy and James Joseph Sylvester, actively used and developed this principle in their research. In the 20th and 21st centuries, the inclusion-exclusion principle continued to be extensively studied and applied in various areas of mathematics. For example, Richard Rado applied the inclusion-exclusion principle in graph theory and combinatorial optimization problems, Gian-Carlo Rota developed generalizations of the principle related to Möbius inversion, and Noga Alon used this principle in counting problems and algorithm analysis.