UPSC Prelims 2008 Q3: Science & Technology – Quantitative Aptitude
In how many different ways can four books A, B, C and D be arranged one above another in a
vertical order such that the books A and B are never in continuous position?
- (a) 9
- (b) 12
- (c) 14
- (d) 18
Answer: (b) 12
Explanation
Arranging four books so two specific books are never in a continuous position
Four books — A, B, C, D — are to be arranged one above another in a vertical order, with the condition that books A and B are never in a continuous (adjacent) position.
| Step | Calculation |
|---|---|
| Total unrestricted vertical arrangements of the 4 books | 4! = 24 |
| Arrangements where books A and B ARE in a continuous position | Treat A-B as one block (2 internal orders: AB or BA), plus the other 2 books = 3 units: 3! × 2 = 12 |
| Arrangements where A and B are NEVER continuous | Total arrangements − continuous-position arrangements = 24 − 12 = 12 |
Why this subtraction method works:
| Point | Detail |
|---|---|
| Total vertical arrangements of the 4 books | 4! = 24 |
| Subtract the 'bad' cases | Arrangements where books A and B are in a continuous position (12 of them) |
Answer: (b) 12.
Why each option is right or wrong
- (a) Incorrect. 9 does not match the correct subtraction calculation.
- (b) CORRECT. 24 total arrangements minus 12 'A-B adjacent' arrangements = 12.
- (c) Incorrect. 14 also does not match the correct calculation.
- (d) Incorrect. 18 significantly overstates the correct count.
Key point
Total arrangements (4! = 24) minus arrangements where A and B ARE adjacent (3! × 2 = 12) gives 12 arrangements where A and B are never adjacent.
Remember
4 books, A and B never adjacent: 24 − 12 = 12 valid arrangements.
Common mistakes
- Forgetting to subtract from the total rather than trying to directly count the 'never adjacent' cases.
- Miscounting the 'adjacent' case by forgetting to multiply by 2 for the AB/BA internal order.
- Confusing this with a different constraint (e.g., 'always adjacent' instead of 'never adjacent').
Exam tip
For 'never adjacent' arrangement problems: Total arrangements − Arrangements where they ARE adjacent = Answer.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)