Permutations
A permutation counts arrangements where order matters. Arranging r items chosen from n distinct items gives nPr = n!/(n−r)! — which is just the counting principle applied to shrinking choices: n for the first slot, n−1 for the second, and so on for r slots. Seating, ranking, forming ordered codes from distinct symbols, and awarding distinct positions (gold/silver/bronze) are all permutations. Two special cases show up often. When some items repeat, divide by the factorial of each repeat count: the arrangements of the letters in LEVEL are 5!/(2!·2!) = 30, because the two L's and two E's are indistinguishable. When items are arranged in a circle, fix one item to kill rotational duplicates, giving (n−1)! arrangements. The GRE keeps n small, so the arithmetic is light — the discipline is recognising that order counts before you reach for a formula.
✅ 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 |