Combinatorics

Abstract

Probabilistic arguments can often be reduced to techniques where one enumerates all possibilities. These techniques are based on the Law of Total Probability and are most useful when there is a set of equally probable basic events and when events of interest consist of combinations of these basic events. In this chapter we focus on such enumerative techniques and derive formal counting techniques collectively called combinatorics. We define four different counting paradigms in Section 3.2,which are expressed in terms of selecting distinguishable balls from an urn or, equivalently, in terms of allocating indistinguishable balls to different urns. To derive expressions for the number of different possibilities for each counting paradigm, we establish recurrence relationships between the number of possibilities in the initial problem to that of a similar, but smaller, problem. This type of argument corresponds to partitioning the counting space into disjoint sets, which are smaller and easier to count.