Multiplication Rule

The multiplication rule is the foundation of all counting techniques.

Basic Rule

If an experiment has $m$ outcomes, and for each outcome a second experiment has $n$ outcomes, then the combined experiment has $m \times n$ outcomes.

Example: A menu has 3 appetizers and 4 main courses.

$$\text{Total combinations} = 3 \times 4 = 12$$

Multi-Stage Experiments

The rule extends to any number of stages:

$$m_1 \times m_2 \times \cdots \times m_k$$

Example: A password has 2 letters followed by 3 digits.

• Letters: $26 \times 26 = 26^2$
• Digits: $10 \times 10 \times 10 = 10^3$
• Total: $26^2 \times 10^3 = 676,000$

Tree Diagrams

Tree diagrams visualize multi-stage experiments:

1. Draw branches for each outcome at stage 1
2. From each branch, draw branches for stage 2
3. Continue for all stages
4. Count the leaf nodes

Example: Coin Flips

For 3 coin flips:

• Stage 1: 2 branches (H, T)
• Stage 2: 2 branches each (4 total)
• Stage 3: 2 branches each (8 total)

$$2 \times 2 \times 2 = 2^3 = 8 \text{ outcomes}$$