UPSC Prelims 2008 Q6: Science & Technology – Quantitative Aptitude
A schoolteacher has to select the maximum possible number of different groups of 3 students
out of a total of 6 students. In how many groups any particular student will be included?
- (a) 6
- (b) 8
- (c) 10
- (d) 12
Answer: (c) 10
Explanation
How many groups of 3 include a particular student
From 6 students, we form groups of 3; we need how many of these groups include one specific student.
| Step | Calculation |
|---|---|
| Fix the particular student | Already in the group, 1 of 3 spots filled |
| Choose remaining 2 members | From the other 5 students |
| Calculation | C(5, 2) = 5×4/2 = 10 |
Why we use C(5,2):
| Point | Detail |
|---|---|
| Particular student | Fixed in every qualifying group |
| Remaining 2 slots | Filled by choosing 2 from the other 5 students |
Answer: (c) 10.
Why each option is right or wrong
- (a) Incorrect. 6 undercounts the correct number of qualifying groups.
- (b) Incorrect. 8 also undercounts the correct total.
- (c) CORRECT. C(5,2) = 10 groups include the particular student.
- (d) Incorrect. 12 overstates the correct count.
Key point
With a particular student fixed in the group, the remaining 2 spots are filled by choosing from the other 5 students: C(5,2) = 10 such groups.
Remember
Groups of 3 from 6 students including one particular student: C(5,2) = 10.
Common mistakes
- Forgetting to fix the particular student first, instead calculating from the full set of 6.
- Using the wrong combination formula (e.g., choosing 3 from 5 instead of 2 from 5).
- Confusing this with the total number of groups possible from all 6 students (C(6,3)=20), rather than those including one specific student.
Exam tip
'How many groups include a specific person' = fix that person, then choose the remaining spots from everyone else.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)