The Counting Principle
The fundamental counting principle says that if one choice can be made in m ways and an independent following choice in n ways, the two together can be made in m × n ways — and this extends to any number of stages. A meal with 4 mains and 3 desserts offers 4 × 3 = 12 combinations; a 3-character code from 26 letters (repeats allowed) offers 26 × 26 × 26 outcomes. The key word is independent: multiply when each stage's options do not depend on the previous choice. When repetition is not allowed, the pool shrinks by one at each stage — a first digit from 5 options, then 4, then 3. And when the task splits into mutually exclusive cases ("the code starts with a vowel OR a digit"), you add the case counts rather than multiply. Deciding multiply-vs-add is the whole skill: multiply across independent stages of one arrangement, add across separate scenarios.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Core counting
| Counting principle | if step 1 has m ways and step 2 has n ways, together m × n ways |
|---|---|
| Factorial | n! = n × (n−1) × … × 2 × 1; 0! = 1 |
| Permutations (order matters) | nPr = n! / (n − r)! |
| Combinations (order does not) | nCr = n! / [r! (n − r)!] |
Handy identities
| Symmetry | nCr = nC(n − r) |
|---|---|
| Choose all or none | nC0 = nCn = 1 |
| Circular arrangements | (n − 1)! ways to seat n around a round table |
| Permutations with repeats | n! / (p! q! …) for repeated items |