Counting sounds like the most elementary thing in mathematics, yet beyond a handful of objects, listing every possibility by hand becomes hopeless. The whole of this chapter exists to count without listing. Two principles do the heavy lifting, and almost every counting question reduces to deciding which one applies.
Multiplication Principle (Fundamental Principle of Counting, FPC): If one task can be done in $m$ ways and, after it is done, a second independent task can be done in $n$ ways, then the two tasks together can be done in $m \times n$ ways. The phrase that signals multiplication is "and then" — a sequence of choices made one after another.
Addition Principle: If a task can be done by method A in $m$ ways or by method B in $n$ ways, and the two methods cannot happen together, then the task can be done in $m + n$ ways. The signal word here is "or" — a choice between mutually exclusive alternatives.
| Situation | Signal word | Principle | Operation |
|---|---|---|---|
| Do step 1, then step 2 | and / then | Multiplication | $m \times n$ |
| Pick option A or option B | or | Addition | $m + n$ |
Factorial. Counting arrangements forces us to multiply long descending strings of integers, so we give that product a name. For a positive integer $n$, $n$ factorial is the product of all positive integers up to $n$:
Factorials grow ferociously fast: $5! = 120$ but $10! = 3{,}628{,}800$. Two facts must be memorised. First, the recursive relation $n! = n \times (n-1)!$, which lets you simplify ratios of factorials without ever expanding them fully. Second, by convention:
Why is $0! = 1$? It is not a fudge. The recursion $n! = n \times (n-1)!$ at $n = 1$ gives $1! = 1 \times 0!$, which forces $0! = 1$. There is also a combinatorial reason: there is exactly one way to arrange nothing — the empty arrangement — so the count of arrangements of zero objects must be $1$, not $0$.
Deeper Insight — one decision tree, two operations: Beginners treat the multiplication and addition principles as separate tricks to memorise, but they are really two readings of the same picture: a decision tree. When choices happen in sequence — first this, then that — the branches multiply, because each early branch splays into a full fan of later ones. When choices are alternatives — this route or that route — the branches add, because you travel down only one of them. The single most reliable habit in counting is to narrate the problem to yourself in plain words and listen for "and" versus "or"; the arithmetic then writes itself. Factorial is simply what the multiplication principle produces when you arrange all of a set's objects in a row — $n$ choices for the first slot, $n-1$ for the next, and so on down to $1$ — which is why it sits at the root of both permutations and combinations.