The Product Rule

If there are m ways to do one thing and n ways to do another, the total number of outcomes is m × n.

Formula: Total = m × n

Example: 3 shirts and 4 trousers: Total outfits = 3 × 4 = 12

Multiple Choices

Extend the rule: just keep multiplying for each additional choice.

Example: 3 starters, 5 mains, 4 desserts: Total meals = 3 × 5 × 4 = 60

Making Lists

You can check your answer by listing pairs (e.g., S1T1, S1T2...), but the product rule is much faster.

Tree Diagrams

Each branch represents a choice. Total outcomes = total branches at the end.

Codes and Passwords

Each position in a code has a number of choices. Multiply them all.

Example: 4-digit PIN (digits 0-9): 10 × 10 × 10 × 10 = 10,000

Without Repetition

If items cannot be reused, the choices reduce by 1 each time.

Example: 3-letter code from 26 letters (no repeats): 26 × 25 × 24 = 15,600

Arranging Items

The number of ways to arrange n distinct items is n!

Formula: n! = n × (n-1) × ... × 1

Example: Ways to arrange 5 books = 5 × 4 × 3 × 2 × 1 = 120

Fixed Positions

If some choices are fixed (e.g. "code must start with A"), that position only has 1 choice.

Example: 4-letter code starting with A: 1 × 26 × 26 × 26