Counting Principles

Fundamental Counting Principle

If one event has m outcomes and another has n outcomes, the total number of combined outcomes is m × n. Extend to any number of independent stages.

Example

Multiply independent choices across stages.

highlighted = computed this step

Step 1 — Choices

List the choices for each stage.

stagechoices132432\begin{array}{cc}\text{stage} & \text{choices} \\ 1 & 3 \\ 2 & 4 \\ 3 & 2\end{array}

Step 2 — First two stages

Multiply the first two stages.

3⋅4=123 \cdot 4 = \hl{12}

Step 3 — All stages

Multiply by the final stage.

12⋅2=2412 \cdot 2 = \hl{24}

Step 4 — Result

State the total number of outcomes.

outcomes=24outcomes= \hl{24}
fundamental-counting-principle Fundamental Counting Principle: Total outcomes = n₁ × n₂ × ⋯ × nₖ where nᵢ is the number of choices at stage i.