CAPF AC 2020 Q117: General Mental Ability – Quantitative Aptitude
The number of ways by which 6 distinct balls can be put in 5 distinct boxes are
- (a) 7776
- (b) 15625
- (c) 720
- (d) 120
Answer: (b) 15625
Explanation
Putting distinct balls into distinct boxes
Given: 6 distinct balls are to be put in 5 distinct boxes.
We need the number of ways.
How to think: ask who is choosing.
Each ball chooses a box.
Every ball has 5 choices, and a box may hold any number of balls.
| Ball | Number of choices |
|---|---|
| Ball 1 | 5 |
| Ball 2 | 5 |
| Ball 3 | 5 |
| Ball 4 | 5 |
| Ball 5 | 5 |
| Ball 6 | 5 |
Total ways = 5 × 5 × 5 × 5 × 5 × 5 = 5⁶ = 15,625.
Where the other options come from:
| Option | What it is |
|---|---|
| 7776 | 6⁵, the powers swapped |
| 15625 | 5⁶, the right count |
| 720 | 6!, arrangements of 6 things in a row |
| 120 | 5!, arrangements of 5 things in a row |
Answer: (b) 15625.
Why each option is right or wrong
- (a) Incorrect. 7776 is 6⁵; it treats the boxes as choosing the balls.
- (b) CORRECT. Each of the 6 balls has 5 choices, so the ways are 5⁶ = 15,625.
- (c) Incorrect. 720 is 6!, which counts arrangements in a line.
- (d) Incorrect. 120 is 5!, which is not the count here.
Key point
n distinct objects into r distinct boxes, no limit on a box: r × r × ... (n times) = rⁿ ways. Here 5⁶ = 15,625.
Remember
n distinct balls into r distinct boxes: r to the power n ways.
Common mistakes
- Swapping the powers and writing 6⁵.
- Using a factorial as if one ball must go in each box.
- Forgetting that a box may be left empty.
Exam tip
Rule: (choices) to the power (choosers). The balls choose, so it is 5 to the power 6.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations (fundamental principle of counting)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)