6.0: Introduction

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If the title of this chapter seems less than inspiring, it’s only because the kind of counting we learned as children was mostly of a straightforward kind. In this chapter, we’re going to learn to answer some more difficult questions like “how many different semester schedules could a college student possibly have?" and “how many different passwords can a customer choose for this e-commerce website?" and “how likely is this network buffer to overflow, given that its packets are addressed to three different destinations?"

The more impressive-sounding name for this topic is combinatorics. In combinatorics, we focus on two tasks: counting things (to find out how many there are), and enumerating things (to systematically list them as individuals). Some things turn out to be hard to count but easy to enumerate, and vice versa.