Counting Methods
Counting is the engine room of probability, and the GRE tests it in its own right too. The whole subject rests on one idea — the counting (multiplication) principle — and two specialisations of it: permutations, where order matters, and combinations, where it does not. The single decision that determines almost every answer is whether arrangement counts: choosing 3 people for a committee is a combination, but seating 3 people in a row is a permutation. The GRE keeps the numbers small enough to reason through, so you rarely need a calculator; what it rewards is setting up the count correctly and, above all, not overcounting. Get the order-matters question right and handle restrictions cleanly, and this becomes one of the more mechanical, dependable topics on the test.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Decide order first: order matters → permutation (nPr); order does not → combination (nCr). This one call fixes most problems.
- Multiply across independent stages of one arrangement; ADD across separate mutually exclusive cases.
- Use symmetry nCr = nC(n−r) to compute the smaller side — 20C18 is just 20C2 = 190.
- For "at least one", count the complement (none) and subtract from the total.
- For "must sit together", glue the items into a block, arrange, then multiply by the block's internal orderings.
- Round-table seatings of n items = (n−1)!, because rotations repeat.
- The GRE keeps n small — a basic calculator is enough; the setup, not the arithmetic, is the challenge.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Using a permutation when order does not matter (or a combination when it does) — the single biggest counting error.
- Multiplying when the scenarios are separate cases that should be added.
- Forgetting to divide out repeats — arrangements of BOOK are 4!/2!, not 4!.
- Ignoring a leading-zero or other restriction and filling positions left-to-right blindly.
- Double-counting a group as several ordered lists (that is exactly what dividing by r! prevents).
- Treating a circular arrangement like a row and using n! instead of (n−1)!.
📈 GRE exam insight & question patterns
Quantitative Comparison — Column A: 6C2. Column B: 6C4. Which is greater?
They are equal by symmetry: 6C2 = 6C4 = 15.
Concept — How many ways can a 4-person committee be chosen from 9 people?
9C4 = 126. Order does not matter, so it is a combination.
Numeric Entry — In how many ways can the letters of the word ERROR be arranged?
5 letters with three R's: 5!/3! = 120/6 = 20.
Multiple-answer — Which situations are permutations (order matters)? (seating in a row / choosing a committee / awarding 1st-2nd-3rd / forming a 3-digit code from distinct digits)
Seating in a row, awarding 1st-2nd-3rd, and forming an ordered code — all care about order. Choosing a committee does not.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
- Everything builds on the counting principle: multiply options across independent stages.
- Order matters → permutation nPr; order does not → combination nCr = nPr divided by r!.
- Add across separate cases; multiply across stages of a single arrangement.
- Divide out repeats — that is the cure for overcounting, and why combinations and circles divide.
- Use nCr = nC(n−r) to compute the easier side.
- For "at least one", subtract the complement from the total.
- For "must sit together", glue into a block, arrange the units, then order within the block.
- The GRE keeps the numbers small — correct setup beats fast arithmetic every time.
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Counting Methods when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
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 |