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Q68 (CAPF/2026) Miscellaneous & General Knowledge › Important Days, Places & Events

A rectangular wall is partitioned into 5 rectangular parts as shown in the following diagram. If you have three different colours to paint the wall and if no two adjacent parts are to be painted with the same colour, in how many different ways can one paint the wall?

Result
Your answer: —  ·  Correct: C

Explanation

To solve this, we identify the adjacencies between the five parts. Let the parts be P1 (top row), P2, P3 (middle row), and P4, P5 (bottom row). According to the diagram, the adjacencies (sharing a boundary line) are:

  • P1 is adjacent to P2 and P3.
  • P2 is adjacent to P1, P3, and P4.
  • P3 is adjacent to P1, P2, and P5.
  • P4 is adjacent to P2 and P5.
  • P5 is adjacent to P3 and P4.

Step 1: Color the triangle formed by P1, P2, and P3. With 3 colors, there are 3 × 2 × 1 = 6 ways.

Step 2: For each way (e.g., P1=Red, P2=Green, P3=Blue), color P4 and P5:

  • P4 must not be Green (adj to P2). Choices for P4: {Red, Blue}.
  • If P4 = Red: P5 cannot be Blue (adj to P3) or Red (adj to P4). Choice: {Green} (1 way).
  • If P4 = Blue: P5 cannot be Blue (adj to P3) or Blue (adj to P4). Choices: {Red, Green} (2 ways).

Total ways = 6 × (1 + 2) = 18.

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