The multiplication rule

What just happened?

Every choice stage branches the tree. To each of the outcomes so far we add $n$ new possibilities — that is why with each stage the number of paths through the tree gets multiplied, rather than merely growing by a few.

The multiplication rule:

If a choice consists of $k$ independent stages, and at stage $i$ there are $n_i$ possibilities, then the number of all outcomes is the product:

When the number of options is the same at every stage ($n_1=\dots=n_k=n$) this simplifies to $N = n^k$.

A worked example:

3 starters, 4 main courses, 2 desserts — how many different dinner sets?

Not $3+4+2=9$ — because each of the 3 starters goes with each of the 4 mains, and each of those 12 pairs with each of the 2 desserts.

When do we ADD? (the addition rule)

We sum only when the cases are mutually exclusive — we carry out one stage “or” the other, not both at once:

A decision shortcut: you combine choices “and–and” → you multiply; you pick “either–or” → you add.

Why this matters for your exams:

The multiplication rule is the foundation of counting: variants of passwords, licence plates, codes, arrangements and permutations. Break the problem into successive independent stages, count the options at each and multiply.

Remember: independent choice stages — the number of outcomes is computed as the product of the options at the successive stages: $N = n_1\cdot n_2\cdots n_k$.