Question

A new flag is to be designed with six vertical stripes using some or all of the colours yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent stripes have the same colour is

• A. \(20 \times 125\)
• B. \(16 \times 192\)
• C. \(12 \times 81\)
• D. \(24 \times 216\)

Hint:

Choose the colour of the first stripe freely, then each next stripe can be any colour except the one just used.

Answer 1

The Correct Option is C

Step 1: First stripe:
The first stripe can be any of the 4 colours: 4 ways.

Step 2: Each later stripe:
Each of the other 5 stripes must differ from the stripe just before it, so it has 3 choices.

Step 3: Multiply:
Number of ways \(= 4 \times 3^5 = 4 \times 243 = 972\).
"Some or all of the colours" is already covered, because the count includes flags that use only 2 or 3 of the colours.

Step 4: Match with the options:
\(12 \times 81 = 972\), \(16 \times 192 = 3072\), \(20 \times 125 = 2500\), \(24 \times 216 = 5184\). Only option (3) equals 972.

Final Answer:
The correct option is (3): \(12 \times 81\).