CAPF AC 2023 Q5: General Mental Ability – Quantitative Aptitude
- (a) 2
- (b) 4
- (c) 6
- (d) 8
Answer: (c) 6
Explanation
Colouring a two-by-two wall
Given: four equal squares in two rows of two; the top-left is green; three colours including green are available; no two adjacent squares may have the same colour; count the colourings.
How to think: adjacent means sharing a side.
The top-right and bottom-left squares touch the green square, so each can take one of the two other colours.
The bottom-right square touches both of them.
| Case | Top-right and bottom-left | Choices for bottom-right | Ways |
|---|---|---|---|
| Same colour | 2 ways (both colour A, or both colour B) | Green or the remaining colour: 2 | 2 × 2 = 4 |
| Different colours | 2 ways (A and B, or B and A) | Only green is left: 1 | 2 × 1 = 2 |
Total: 4 + 2 = 6.
Answer: (c) 6.
Why each option is right or wrong
- (a) Incorrect. 2 counts only the case where the two middle squares differ.
- (b) Incorrect. 4 counts only the case where the two middle squares match.
- (c) CORRECT. 4 + 2 = 6 colourings.
- (d) Incorrect. 8 treats the last square as always having 2 choices.
Key point
Grid colouring: fix the given square, count choices for its neighbours, then split into cases by whether those neighbours match. Diagonal squares are not adjacent, so they may share a colour.
Remember
Split by whether the two neighbours match: 4 + 2 = 6.
Common mistakes
- Treating diagonal squares as adjacent and banning green in the bottom-right.
- Giving the last square 2 choices in every case and answering 8.
- Forgetting that the top-left colour is already fixed as green.
Exam tip
Educated guess: two squares touch green and each has 2 choices, giving 4 starts. The last square has 1 or 2 choices, so the total lies between 4 and 8. Only 6 is in between.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations (fundamental principle of counting)
- R. S. Aggarwal, A Modern Approach to Verbal and Non-Verbal Reasoning — Logical puzzles