Product rule

Description

The product rule for counting (also called the multiplication principle) states: if one event can happen in $m$ ways and a second event can happen in $n$ ways, the total number of ways both can happen is $m \times n$.

$$\text{Total outcomes} = m \times n \times \ldots$$

Example: A menu has 4 starters, 5 mains and 3 desserts. How many different three-course meals are possible? $$4 \times 5 \times 3 = 60 \text{ meals}$$

Example: How many 3-digit codes can be made from digits 1–9 if digits cannot be repeated? $$9 \times 8 \times 7 = 504$$

The rule extends to as many events as needed. When repetition is not allowed, reduce the number of choices by 1 for each subsequent choice made.

Common error: adding the number of choices rather than multiplying, or forgetting to reduce the count when repetition is not allowed.

Links

Theoretical probabilities Probability trees