The flag of a new political party is to be designed with six vertical strips…
2025
The flag of a new political party is to be designed with six vertical strips using some or all of the colours using yellow, green, blue and red. Then, the number of ways this can be made such that no two adjacent strips have the same colour is:
- A.
1024
- B.
812
- C.
972
- D.
1260
Show answer & explanation
Correct answer: C
Concept: This is an application of the fundamental counting principle for a sequence of choices where each choice is restricted only by the choice immediately before it (not by every earlier choice) - multiply the number of options available at each step to get the total count of the sequence.
Application:
The first stripe can be any of the 4 available colours, so there are 4 choices for it.
Each of the remaining five stripes must differ only from the stripe immediately before it, so each of these five stripes has 3 valid colour choices.
By the counting principle, the total number of colourings is the product of the choices at every stripe: 4 × 3 × 3 × 3 × 3 × 3 = 4 × 35.
4 × 35 = 4 × 243 = 972.
Cross-check: Grouping the factors differently confirms the same product — (4 × 3) × 34 = 12 × 81 = 972 — so the total is consistent no matter how the multiplication is grouped.
Hence, the required number of ways is 972.