Counting Things: Combinatorics

Abstract

Up to now, our work has been almost entirely qualitative. The concepts of a set, relation, function, recursion and induction are, in themselves, non-numerical although they have important numerical applications as, for example, sets of integers or recursive definitions on the natural numbers. In this chapter we turn to directly quantitative matters, and specifically to problems of counting. We tackle two topics: rules for determining the number of elements of a large set from the number of elements of smaller sets, rules for calculating the number of possible selections of k items out of a set with n elements.