Uma interpretação combinatória, via ladrilhamento para a sequência de Fibonacci

Abstract

Mathematics students who have already gone through the subjects of combinatorial analysis have seen tests that use direct and formalistic counting involving the coefficient. Introduction 9 binomial. the definition of

no k ! , that is, the number of ways to choose k elements from a set of n elements, provides a tool that is used to demonstrate many binomial properties. Per on the other hand, it can be interpreted by its combinatorial aspect for the demonstrations of identities, arising from their meaning. For example, to prove identity k

no k ! = n

n − 1 k − 1 ! , suppose the following problem: “the number of ways to form teams with k people from a group of n people, in which one of these k people is the captain”. For that we will count such a set in two different ways. First choosing the team and then choosing one of these k people to be captain. Another way would be to choose a captain out of n people, and then choose k − 1 of the remaining n − 1 people to get the rest of the team. So the number of ways to choose first the team and then choosing a captain from the team members (the left side of the equation) is the same as the number of ways to choose one captain first and then choose the rest of the team (the right side of the equation). Consequently, proving that identity is equivalent to simply consider a real world example. The resulting proof is much more satisfactory and accessible than algebraic manipulations of the formula no k

= no! k!(n−k)! .