UPSC Prelims 2007 Q134: Science & Technology – Quantitative Aptitude
In how many maximum different ways can 3 identical balls be placed in 12 squares (each
ball is placed in the exact centre of a square) shown in the figure above such that they do not lie
along the same straight line?
- (a) 144
- (b) 200
- (c) 204
- (d) 216
Answer: (b) 200
Explanation
3 balls in 12 squares — not all in one straight line
The 12 squares form a 3 × 4 grid (3 rows, 4 columns).
Step 1 — All ways to choose 3 squares:
C(12, 3) = 220.
Step 2 — Subtract the ways where all 3 lie on one straight line:
| Line type | Count |
|---|---|
| Rows (3 rows of 4 squares → C(4,3) = 4 each) | 3 × 4 = 12 |
| Columns (4 columns of 3 squares → 1 each) | 4 |
| Diagonals of 3 squares (2 in each direction) | 4 |
| Total in a line | 20 |
Step 3: 220 − 20 = 200.
So the maximum number of different ways to place 3 identical balls in the exact centre of 3 squares, with no three on one straight line, is 200.
Answer: (b) 200.
Why each option is right or wrong
- (a) Incorrect. 144 subtracts far too many cases.
- (b) CORRECT. C(12,3) − 20 collinear sets = 220 − 20 = 200.
- (c) Incorrect. 204 forgets the 4 diagonal lines.
- (d) Incorrect. 216 subtracts only the 4 columns and misses rows and diagonals.
Key point
Total C(12,3) = 220. Collinear: rows 12 + columns 4 + diagonals 4 = 20. Answer 200.
Remember
220 total − 20 straight-line sets = 200.
Common mistakes
- Forgetting the diagonals → 204.
- Counting a row of 4 as just 1 line — a row of 4 gives C(4,3) = 4 triples.
- Counting diagonals of length 2 — you need 3 squares in a line.
Exam tip
'Not in a line' = Total − in a line. List lines by type: rows, columns, diagonals.
Sources
- NCERT Class XI — Mathematics, Ch 6 (Permutations and Combinations)
- Arun Sharma — How to Prepare for Quantitative Aptitude for the CAT (Permutations and Combinations)