Arrangements
An arrangement counts ordered outcomes, so swapping two items gives a new count. The fundamental counting principle drives everything: if step 1 can happen in m ways and step 2 in n ways, the pair happens in m × n ways. Arranging all n distinct objects in a row gives n!; arranging only r of them gives nPr = n!/(n−r)!. The CAT favourite is the digit/number problem — "how many 4-digit numbers from these digits" — where you fill positions left to right and watch two traps: the leading digit cannot be 0, and "no repetition" versus "repetition allowed" change the count completely. With repetition allowed and k choices per slot, r slots give kʳ. A fast habit: draw r blanks, write the count of valid choices in each, and multiply.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Core counting
| Permutations (order matters) | nPr = n! / (n − r)! |
|---|---|
| Combinations (order ignored) | nCr = n! / (r!(n − r)!) |
| Arrangements with repetition | n! / (p! q! r! …) for repeated items |
| Fundamental counting principle | total = (ways for step 1) × (ways for step 2) × … |
| Symmetry of combinations | nCr = nC(n − r) |
Circular & special
| Circular arrangement (n distinct) | (n − 1)! |
|---|---|
| Necklace / garland (reflections same) | (n − 1)! / 2 |
| Sum of all combinations | nC0 + nC1 + … + nCn = 2ⁿ |
| At least one selection (from n) | 2ⁿ − 1 |
| Relation P and C | nPr = nCr × r! |