An example of optimization of a combinatorial calculus

Abstract

In this paper, we initially recall a tool of calculus, the backwards differences, and we show how it allows to obtain some useful relations. We make a first application and we find a recursive relation for . Afterwards we make a second application and we prove the relation It allows to prove the relation ,with p + q =1. Finally, we show how to optimize the previous calculus and to obtain the latter relation in a easier way by means of the distribution of the number of goals scored in a table-football game.