Counting

This chapter will be devoted to problems involving counting. Of course, everybody knows how to count, but sometimes this can be quite a tricky business. Consider, for example, the following questions:

1. (i) In how many different ways can seven identical objects be arranged in a row?

2. (ii) In how many different ways can a group of three ball bearings be selected from a bag containing eight?

Problems of this type are called combinatorial. If you try to solve them directly by counting all the possible alternatives, you will find this to be a laborious and time-consuming procedure. Fortunately, a number of clever tricks are available which save you from having to do this. The branch of mathematics which develops these is called combinatorics and the purpose of the present chapter is to give a brief introduction to this topic.

A fundamental concept both in this chapter and the subsequent ones on probability theory proper will be that of an ‘experience’ which can result in several possible ‘outcomes’. Examples of such experiences are:

1. (a) throwing a die where the possible outcomes are the six faces which can appear,

2. (b) queueing at a bus-stop where the outcomes consist of the nine different buses, serving different routes, which stop there.

If A and B are two separate experiences, we write A ∘ B to denote the combined experience of A followed by B.