UPSC Prelims 2007 Q136: Science & Technology – Quantitative Aptitude
Amit has five friends: 3 girls and 2 boys. Amit’s wife also has five friends: 3 boys and 2 girls.
In how many maximum different ways can they invite 2 boys and 2 girls such that two of them
are Amit’s friends and two are his wife’s friends?
- (a) 24
- (b) 38
- (c) 46
- (d) 48
Answer: (c) 46
Explanation
Inviting 2 boys + 2 girls: 2 from Amit's friends, 2 from his wife's friends
| Group | Girls | Boys |
|---|---|---|
| Amit's friends | 3 | 2 |
| Wife's friends | 2 | 3 |
Three possible splits:
| Case | From Amit | From wife | Ways |
|---|---|---|---|
| 1 | 2 girls | 2 boys | C(3,2) × C(3,2) = 3 × 3 = 9 |
| 2 | 2 boys | 2 girls | C(2,2) × C(2,2) = 1 × 1 = 1 |
| 3 | 1 boy + 1 girl | 1 boy + 1 girl | (2 × 3) × (3 × 2) = 6 × 6 = 36 |
| Total | Cases 1-3 | Added together | 46 |
Answer: (c) 46.
Why each option is right or wrong
- (a) Incorrect. 24 misses whole cases.
- (b) Incorrect. 38 under-counts case 3.
- (c) CORRECT. 9 + 1 + 36 = 46 ways across the three splits.
- (d) Incorrect. 48 over-counts; the three cases add to 46.
Key point
Split into cases (2G+2B / 2B+2G / 1B1G+1B1G): 9 + 1 + 36 = 46.
Remember
Split into cases, then add: 9 + 1 + 36 = 46 ways.
Common mistakes
- Forgetting the mixed case (1 boy + 1 girl from each side) — the largest case.
- Mixing up who has more boys — Amit has 2 boys, his wife has 3 boys.
- Multiplying where you should add — add across cases, multiply within a case.
Exam tip
Combinations rule: within a case → multiply; across cases → add.
Sources
- NCERT Class XI — Mathematics, Ch 6 (Permutations and Combinations)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)