Introduction

Counting is one of the very first mathematical skills we learn, and as the objects to be counted grow large, simply counting them one by one stops being practical. This section introduces two powerful counting techniques — permutations and combinations — that let us count large collections of possibilities WITHOUT having to list or enumerate every single one.

The key distinction between the two ideas, stated right at the start, is about ORDER. A permutation is about the number of different ARRANGEMENTS of a set of objects, where the order in which the objects are placed genuinely matters — swapping two objects gives a different, distinct arrangement. A combination is about the number of different SELECTIONS from a set of objects, where the order doesn't matter at all — only which objects end up chosen is important, not the sequence they were picked in.

Two motivating examples open the chapter. The first is a smartphone passcode: a passcode is exactly 4 of the 10 available digits, and — crucially — the ORDER of these four digits cannot be changed for the passcode to still work, so this is fundamentally a permutation-style question. Reasoning it out: the first digit position can be any of the 10 digits, the second position can independently also be any of the 10 digits (a digit is allowed to repeat, since nothing says otherwise), and likewise for the third and fourth positions. Multiplying these four independent choices together gives $10\times10\times10\times10=10{,}000$ distinct possible passcodes — a number that would be essentially impossible to reach by writing every passcode out by hand, which is exactly the point: the multiplication shortcut replaces a hopeless listing task with a two-line calculation.

The second motivating example is harder and is deliberately left unsolved here, to be returned to once the chapter's tools are built up: a school cricket team of eleven players is to be photographed together with the principal and the two vice-principals, using seven chairs arranged in a row — three reserved for the principal and vice-principals, four more for players sitting, and the remaining seven players standing behind the chairs. The question, 'in how many different ways can the eleven players take their positions?', previews the kind of multi-stage arrangement problem (combining a SELECTION of who sits where with a PERMUTATION of how they're arranged) that later sections of the chapter equip us to answer cleanly. The introduction closes by inviting the reader to attempt this on their own first, before the formal tools are developed.