UPSC Prelims 2004 Q110: Science & Technology – Quantitative Aptitude
Nine different letters are to be dropped in three different letter boxes. In how many
different ways can this be done?
- (a) 27
- (b) 3^9
- (c) 9^3
- (d) 3^9 − 3
Answer: (b) 3^9
Explanation
Posting nine letters in three letter boxes
| Step | Count |
|---|---|
| Each letter | Can go into any of the 3 boxes |
| Nine different letters | 3 x 3 x ... x 3 (nine times) |
| Total ways | 3^9 |
Each of the nine different letters can be dropped into any one of the three boxes independently, so the number of ways is 3^9.
Answer: (b) 3^9.
Why each option is right or wrong
- (a) Incorrect. 27 is only 3^3, not 3^9.
- (b) CORRECT. 3 choices for each of 9 letters gives 3^9.
- (c) Incorrect. 9^3 would be 9 choices for each of 3 boxes.
- (d) Incorrect. 3^9 - 3 wrongly removes all-in-one-box cases.
Key point
Objects into boxes: choices per object raised to the number of objects.
Remember
Nine letters into three boxes can be done in 3^9 ways.
Common mistakes
- Using 9^3 by reversing objects and boxes.
- Subtracting cases where all letters go in one box.
- Using 9 x 3 = 27 instead of the power 3^9.
Exam tip
Distinct objects, distinct boxes: (number of boxes)^(number of objects).
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)