Fundamental Principle of Counting
The fundamental principle of counting is the master rule: if one task can be done in m ways and a second, independent task in n ways, then both together can be done in m × n ways. The key word is independent — the number of ways to do the second task must not depend on which option you chose for the first. The cleanest way to apply it in CAT is the "slots" method: draw a blank for each decision, write the number of choices in each blank, and multiply. For a 3-course meal with 4 starters, 5 mains and 3 desserts, that is 4 × 5 × 3 = 60 meals. The principle scales to any number of stages: just keep multiplying the choice-counts. Whenever you can describe a configuration as "first do this, then this, then this", you are in multiplication territory, and you almost never need a formula — just careful slot-filling.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
The two principles
| Multiplication principle (AND) | If stage 1 has m ways and stage 2 has n ways, the task has m × n ways |
|---|---|
| Addition principle (OR, exclusive) | If a task is done by method A (m ways) OR method B (n ways), disjoint, total = m + n |
| k independent stages | n₁ × n₂ × … × n_k ways |
| Choices each from r boxes (repetition allowed) | nʳ (n options, r positions) |
Standard counting models
| Functions from A (m elements) to B (n elements) | nᵐ |
|---|---|
| r-digit numbers, no leading zero, repetition allowed | 9 × 10^(r−1) |
| Subsets of an n-element set | 2ⁿ |
| At-least-one | (total arrangements) − (arrangements with none) |
| r-letter strings from an n-letter alphabet | nʳ |