Multiplication Principle
The multiplication principle is the fundamental principle applied to sequential decisions, and it is where most CAT counting questions actually live. The phrase to listen for is "AND" — you must do this AND this AND this. Two refinements matter at the CAT level. First, watch for restrictions that change the count at a later slot: in a number-plate or password problem, "no repeated digit" means the second slot has one fewer choice than the first. Second, fill the most constrained slot first. If a 3-digit number must be even and have no leading zero, decide the units digit (must be even) and the leading digit (no zero) before the free middle slot, so the constraints do not collide. Functions, passwords, number plates and digit problems are all multiplication-principle questions in disguise: count the choices position by position, respect each constraint as you reach its slot, and multiply.
✅ 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 |