3 shirts, 4 trousers, 2 pairs of shoes. How many outfits? Not 9. Not 24 by luck — by a rule: 3 × 4 × 2 = 24. The multiplication principle is the foundation of all counting.
The fundamental counting principle: if one choice can be made in m ways and a second independent choice in n ways, the two together can be made in m × n ways. Extends to any number of stages.
Password spaces, menu combos, license plates, network addresses, probability denominators — almost every 'how many possibilities?' question starts here.
A menu has 4 starters, 6 mains, 3 desserts. How many 3-course meals?
Works only when the choices are independent — each stage's options don't depend on earlier picks.
A PIN is 4 digits, each 0–9, repeats allowed. How many PINs?
How many license plates with 3 letters then 3 digits?
26 × 26 × 26 × 10 × 10 × 10 = 26³ × 10³ = 17,576 × 1,000 = 17,576,000.
Add for 'or', multiply for 'and'. 'A starter OR a dessert' → 4 + 3 = 7. 'A starter AND a dessert' → 4 × 3 = 12. Mixing these up is the classic slip.
If choices *aren't* independent (no repeats allowed, say), the later counts shrink: 10 × 9 × 8 × 7 for a no-repeat 4-digit code. That's a permutation.
- Independent stages → multiply the options at each stage.
- 'AND' → multiply; 'OR' → add.
- Foundation of permutations, combinations, and probability denominators.