UPSC Prelims 2004 Q139: Science & Technology – Quantitative Aptitude
In how many different ways can six players be arranged in a line such that two of them, Ajit
and Mukherjee, are never together?
- (a) 120
- (b) 240
- (c) 360
- (d) 480
Answer: (d) 480
Explanation
Arranging six players with two never together
| Step | Count |
|---|---|
| All arrangements of six players | 6! = 720 |
| Ajit and Mukherjee together | Treat them as one unit: 5! x 2 = 240 |
| Never together | 720 - 240 = 480 |
The number of arrangements with Ajit and Mukherjee never together is 720 - 240 = 480.
Answer: (d) 480.
Why each option is right or wrong
- (a) Incorrect. 120 is only 5! and ignores the order of the two.
- (b) Incorrect. 240 is the number of arrangements with them together.
- (c) Incorrect. 360 is half of 720 and has no basis here.
- (d) CORRECT. 720 - 240 = 480 ways.
Key point
Not together = total minus together, where together uses a unit x 2.
Remember
Six players can be arranged in 480 ways with two never adjacent.
Common mistakes
- Forgetting the 2! order of the pair.
- Giving the number of arrangements with them together.
- Dividing 720 by 2 for the condition.
Exam tip
Use the complement method for never together.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)