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 so that no two adjacent stripes have the same colour is

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

Hint:

When arranging objects with restrictions, use the multiplication principle and subtract choices for the restricted conditions.

Answer 1

The Correct Option is C

This is a problem of arranging 6 stripes where no two adjacent stripes can have the same colour. There are 4 colours available. For the first stripe, we have 4 choices, for the second stripe, we have 3 choices (since it can't be the same colour as the first), and for each of the remaining 4 stripes, we again have 3 choices. Thus, the total number of ways to design the flag is: \[ 4 \times 3 \times 3 \times 3 \times 3 \times 3 = 4 \times 3^5 = 20 \times 125 \] Therefore, the Correct Answer is \( 20 \times 125 \). \[ \boxed{20 \times 125} \]