How to Solve Combinations (GRE/GMAT)

How to Solve Combinations (GRE/GMAT)

Score Impact: Combinations appear 4-6 times per GRE/GMAT Quant section—mastering them can boost your score by 30-50 points by eliminating careless errors and saving 30+ seconds per question.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The exam isn’t testing your ability to compute factorials—it’s testing: 1. Precision in reading conditions (e.g., "without replacement," "order doesn’t matter"). 2. Decision-making under constraints (e.g., "at least one," "no two from the same group"). 3. Avoiding overcounting/undercounting (e.g., treating identical items as distinct).

ANATOMY OF THE QUESTION

Structure Breakdown

Representative Example

Question: A company has 6 employees: 4 engineers and 2 designers. If a team of 3 is selected at random, what is the probability that the team includes exactly 1 designer?

Breakdown: - Stem: Team of 3 from 6 employees (4E, 2D). - Condition: Exactly 1 designer (so 2 engineers). - Answer Choices: (A) 1/5 (B) 2/5 (C) 3/5 (D) 4/5 (E) 1/2

THE DECISION FRAMEWORK (Step-by-Step)

Run this every time—no exceptions.

1. Identify the total pool and subgroups.
2. Total items = ?

3. Subgroups (e.g., engineers/designers, men/women) = ?

4. Determine if order matters.

5. If order matters → Permutation (P(n,r)).

6. If order doesn’t matter → Combination (C(n,r)).

7. Check for restrictions.

8. "At least one" → Use complementary counting (Total – Unwanted).

9. "No two from the same group" → Multiply combinations.

10. Calculate the numerator (desired outcomes).

11. Break into cases if needed (e.g., 1D + 2E, 2D + 1E).

12. Calculate the denominator (total possible outcomes).

13. Total ways to choose the group (e.g., C(6,3)).

14. Simplify and match to answer choices.

15. Reduce fractions early to avoid calculation errors.

Worked Examples

Example 1 – Straightforward

Question: A bag contains 5 red marbles and 3 blue marbles. If 2 marbles are drawn at random, what is the probability that both are red?

Framework Application: 1. Total pool: 8 marbles (5R, 3B). 2. Order doesn’t matter → Combination. 3. No restrictions (just "both red"). 4. Numerator: C(5,2) = 10 (ways to choose 2 red). 5. Denominator: C(8,2) = 28 (total ways to choose 2 marbles). 6. Probability: 10/28 = 5/14.

Answer: 5/14 (not listed—recheck: C(5,2)=10, C(8,2)=28 → 5/14).

Elimination: - (A) 1/4 (too low), (B) 5/14 (correct), (C) 1/2 (too high), (D) 3/4 (way too high), (E) 1/7 (too low).

Example 2 – Common Trap (Ignoring Conditions)

Question: A committee of 4 is chosen from 6 men and 4 women. What is the probability that the committee has at least 1 woman?

Trap: Students calculate C(4,1)C(6,3) + C(4,2)C(6,2) + ... (tedious).

Correct Approach: 1. Total pool: 10 people (6M, 4W). 2. Order doesn’t matter → Combination. 3. "At least 1 woman" → Use complementary counting.
- Total ways: C(10,4) = 210.
- Unwanted (0 women): C(6,4) = 15.
- Desired: 210 – 15 = 195. 4. Probability: 195/210 = 13/14.

Answer: 13/14 (not listed—recheck: C(10,4)=210, C(6,4)=15 → 195/210=13/14).

Elimination: - (A) 1/2 (too low), (B) 5/6 (close but wrong), (C) 13/14 (correct), (D) 14/15 (too high), (E) 1/3 (way off).

Example 3 – Hard Variant (Multiple Cases)

Question: A team of 5 is chosen from 7 seniors and 5 juniors. What is the probability that the team has more seniors than juniors?

Framework Application: 1. Total pool: 12 people (7S, 5J). 2. Order doesn’t matter → Combination. 3. "More seniors than juniors" → Cases:
- 3S + 2J
- 4S + 1J
- 5S + 0J 4. Calculate each case:
- C(7,3)C(5,2) = 3510 = 350
- C(7,4)C(5,1) = 355 = 175
- C(7,5)C(5,0) = 211 = 21
- Total desired: 350 + 175 + 21 = 546 5. Denominator: C(12,5) = 792. 6. Probability: 546/792 = 91/132.

Answer: 91/132.

Elimination: - (A) 1/2 (too simple), (B) 7/12 (close but wrong), (C) 91/132 (correct), (D) 5/6 (too high), (E) 1/3 (too low).

WRONG ANSWER PATTERNS

Common Mistakes

TIME STRATEGY

• Target time: 1:30 per question.
• When to skip: If stuck after 1 minute, flag and return.
• Minimum work:
• Identify total pool and subgroups.
• Write C(n,r) for numerator/denominator.
• Simplify before calculating.

BACKSOLVING AND SHORTCUTS

1. Plug in numbers: For small n, list possibilities (e.g., C(4,2) = 6 → list AB, AC, AD, BC, BD, CD).
2. Symmetry: If "at least 1" is asked, use 1 – P(none).
3. Elimination: Cross out answers that violate conditions (e.g., probability >1).
4. Avoid factorials: Use C(n,r) = C(n,n-r) to simplify (e.g., C(10,7) = C(10,3)).

1-Minute Recap

"Here’s the deal: Combinations test your ability to read carefully, not your math skills. Every time, ask: 1. Order or no order? If no, use C(n,r). 2. Any restrictions? Break into cases or use complements. 3. Calculate numerator and denominator separately, then simplify. 4. Eliminate traps—answers that ignore conditions or overcount. Spend 30 seconds setting up, 30 seconds calculating, and 30 seconds checking. If you’re stuck, flag it and move on. This is a high-value question—nail it, and you’re 30 points closer to your target score."

Final Note: Under timed conditions, write down C(n,r) formulas immediately. Simplify early. And always ask: "Does this answer make sense?"