Definition
Product rule for counting
Multiply the choices available at each stage to count all possible complete outcomes.
What it means
Imagine building one complete choice by taking one option from each row. If there are 4 badge shapes and 3 colours, every shape can pair with every colour. Four groups of three make 12 different badges, so multiplication counts the pairings without writing every one down.
Remember it
Choose from this row AND that row: multiply the row counts.
Example
A game avatar needs one of 3 hats, one of 2 capes, and one of 4 badges. Every hat can go with every cape and badge, so 3 × 2 × 4 = 24 complete avatars.
Watch for
Do not keep multiplying unchanged stage counts when a rule removes choices. Recount the options on each branch or split the problem into cases.
Common mix-up
Add counts for separate either/or cases; multiply counts when a complete outcome needs one choice and then another.
Exact meaning
For successive stages, when every counted choice at one stage can be followed by every counted choice at the next, the number of complete outcomes equals the product of the stage counts. With more stages, multiply all stage counts; when options change by branch, count each branch separately.