Multiplication principle

What the formula says

N = n₁ · n₂ · n₃

Three shirts and four pairs of trousers make 3 · 4 = 12 outfits: each shirt can be worn with each pair of trousers. Add two pairs of shoes and every one of those twelve outfits can be finished in two ways, so 12 · 2 = 24. The rule holds when the options in one step do not depend on what was chosen in the earlier ones. With only two steps, set the third to 1.

Worked examples

A menu. Two starters, five mains and three desserts: 2 · 5 · 3 = 30 different three-course meals.

A code. A lock with three wheels of ten digits each: 10 · 10 · 10 = 1,000 codes.

Backwards. 60 outcomes from 5 options in one step and 3 in another leaves 60 / (5 · 3) = 4 options in the last step.

Common mistakes

• Adding instead of multiplying. 3 shirts and 4 pairs of trousers is 12 outfits, not 7.
• Forgetting that options shrink. If an item cannot be chosen twice, the later steps have fewer options; that is permutations, not this rule.