UPSC Prelims 2007 Q140: Science & Technology – Quantitative Aptitude
Groups, each containing 3 boys, are to be formed out of 5 boys - A, B, C, D and E - such that
no one group contains both C and D together. What is the maximum number of different groups?
- (a) 5
- (b) 6
- (c) 7
- (d) 8
Answer: (c) 7
Explanation
Groups of 3 from 5 boys — C and D never together
| Step | Calculation | Value |
|---|---|---|
| All groups of 3 from 5 | C(5, 3) | 10 |
| Groups with both C and D | Third member from A, B, E → C(3, 1) | 3 |
| Allowed groups | 10 − 3 | 7 |
List them: ABC, ABD, ABE, ACE, ADE, BCE, BDE → 7.
Shortcut: groups with exactly one of C or D: 2 × C(3, 2) = 6; groups with neither: C(3, 3) = 1; total 7.
Answer: (c) 7.
Why each option is right or wrong
- (a) Incorrect. 5 removes too many groups.
- (b) Incorrect. 6 misses one valid group.
- (c) CORRECT. 10 − 3 = 7 groups.
- (d) Incorrect. 8 includes at least one group with both C and D.
Key point
Total C(5,3) = 10; groups with C and D together = 3; answer = 7.
Remember
10 groups − 3 with C and D = 7.
Common mistakes
- Removing groups that contain C or D, instead of C and D.
- Forgetting that C and D together still need a third member (3 choices).
- Not listing the groups to double-check.
Exam tip
'Never together' = Total − Together. With only 10 groups, listing them is also quick.
Sources
- NCERT Class XI — Mathematics, Ch 6 (Permutations and Combinations)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)