The Multiplication Rule (Rule of Product)
Combinatorics — Grade 11
📐 Definition
If one action can be performed in $n$ ways, and for each of those ways a second action can be performed in $m$ ways, then the two actions together can be performed in
$$n \cdot m \text{ ways.}$$
This is called the Multiplication Rule (or Rule of Product).
🔗 Extension to Multiple Steps
The rule extends naturally to any number of sequential, independent choices:
If a procedure consists of $k$ stages, where
- Stage 1 can be done in $n_1$ ways,
- Stage 2 can be done in $n_2$ ways (for each outcome of Stage 1),
- Stage 3 can be done in $n_3$ ways, …
- Stage $k$ can be done in $n_k$ ways,
then the total number of outcomes is:
$$n_1 \cdot n_2 \cdot n_3 \cdot \ldots \cdot n_k$$