UPSC Prelims 2005 Q111: Science & Technology – Quantitative Aptitude
A square is divided into 9 identical smaller squares. Six identical balls are to be placed in
these smaller squares such that each of the three rows gets at least one ball (one ball in one
square only). In how many different ways can this be done?
- (a) 27
- (b) 36
- (c) 54
- (d) 81
Answer: (d) 81
Explanation
Placing six balls in a 3 x 3 grid with each row non-empty
| Step | Count |
|---|---|
| All ways to put 6 balls in 9 squares | C(9,6) = 84 |
| Ways with one particular row empty | 6 balls in the other 6 squares = 1 |
| Three rows, so subtract | 84 - 3 = 81 |
Six balls in nine squares can be placed in 84 ways, and the 3 arrangements where a whole row is empty are removed, giving 81.
Answer: (d) 81.
Why each option is right or wrong
- (a) Incorrect. 27 is far below the count of valid placements.
- (b) Incorrect. 36 does not follow from C(9,6).
- (c) Incorrect. 54 is not 84 minus the empty-row cases.
- (d) CORRECT. 84 - 3 = 81 placements.
Key point
Count all selections, then subtract the cases that violate the rule.
Remember
C(9,6) - 3 = 81 placements.
Common mistakes
- Using permutations for identical balls.
- Subtracting only one empty-row case.
- Forgetting that two rows cannot both be empty.
Exam tip
Identical objects in distinct cells: use combinations.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)