UPSC Prelims 2005 Q103: Science & Technology – Quantitative Aptitude
2 men and 1 woman board a bus in which 5 seats are vacant. One of these five seats is
reserved for ladies. A woman may or may not sit on the seat reserved for ladies but a man can
not sit on the seat reserved for ladies. In how many different ways can the five seats be occupied
by these three passengers?
- (a) 15
- (b) 36
- (c) 48
- (d) 60
Answer: (b) 36
Explanation
Seating two men and one woman on five seats
| Case | Number of ways |
|---|---|
| Woman on the ladies' seat | Two men on the 4 general seats: 4 x 3 = 12 |
| Woman on a general seat (4 choices) | Men on the 3 remaining general seats: 4 x (3 x 2) = 24 |
| Total | 12 + 24 = 36 |
A man cannot sit on the reserved seat, so it is either taken by the woman or left empty. The total is 36 arrangements.
Answer: (b) 36.
Why each option is right or wrong
- (a) Incorrect. 15 undercounts the arrangements.
- (b) CORRECT. 12 + 24 = 36 seatings.
- (c) Incorrect. 48 allows men on the reserved seat.
- (d) Incorrect. 60 is 5P3 and ignores the restriction.
Key point
Handle the restricted seat first: woman on it or not.
Remember
With the reserved seat rule, 2 men and 1 woman can be seated in 36 ways.
Common mistakes
- Using 5P3 = 60 and ignoring the reservation.
- Letting a man take the ladies' seat.
- Forgetting that the ladies' seat can stay empty.
Exam tip
Split into cases by who occupies the restricted seat.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)