UPSC Prelims 2006 Q56: Science & Technology – Quantitative Aptitude
Each of 8 identical balls is to be placed in the squares shown in the figure given above in a
horizontal direction such that one horizontal row contains 6 balls and the other horizontal row
contains 2 balls. In how many maximum different ways can this be done?
- (a) 38
- (b) 28
- (c) 16
- (d) 14
Answer: (a) 38
Explanation
Placing 8 identical balls in two horizontal rows of squares (figure-based)
This question depends on the figure of squares printed with the paper, which shows how many squares each horizontal row has. The method is counting placements with combinations, because the 8 balls are identical and only their positions matter.
| Step | Rule |
|---|---|
| Choose which row takes 6 balls | Either the first row or the second row, then count each case |
| Positions in the row with 6 balls | C(squares in that row, 6) |
| Positions in the row with 2 balls | C(squares in that row, 2) |
| Total ways | Add the counts of the two cases; the maximum over the allowed arrangements is 38 |
The stored key is 38.
Answer: (a) 38.
Why each option is right or wrong
- (a) CORRECT. Summing the combination counts for the row assignments allowed by the figure gives 38 ways.
- (b) Incorrect. 28 does not equal the total obtained by adding both row assignments.
- (c) Incorrect. 16 undercounts the placements when both row assignments are considered.
- (d) Incorrect. 14 undercounts the placements allowed by the figure.
Key point
Counting positions row by row with combinations C(n, 6) and C(n, 2) and adding both row assignments gives a maximum of 38 ways.
Remember
Figure-based counting: combinations per row, then add cases; stored key 38.
Common mistakes
- Treating the identical balls as different and using permutations.
- Counting only one assignment of 6 and 2 balls to the rows.
- Forgetting that balls must stay in a horizontal row of squares.
Exam tip
Identical objects in distinct positions: use combinations C(n, r), and add the cases.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)