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

Options

A. `12 xx 81`
B. `16 xx 192`
C. `20 xx 125`
D. `24 xx 216`

Answer

To solve the problem of designing a flag with six vertical stripes using the colors yellow, green, blue, and red, while ensuring that no two adjacent stripes have the same color, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number of Colors**: We have 4 colors available: yellow, green, blue, and red.

2. **Choose the Color for the First Stripe**: For the first stripe, we can choose any of the 4 colors. Therefore, there are 4 options for the first stripe.

\[ \text{Choices for the first stripe} = 4 \]

3. **Choose the Color for the Second Stripe**: The second stripe cannot be the same color as the first stripe. Hence, we have 3 remaining colors to choose from.

\[ \text{Choices for the second stripe} = 3 \]

4. **Choose the Color for the Third Stripe**: Similarly, the third stripe cannot be the same color as the second stripe. Thus, we again have 3 choices.

\[ \text{Choices for the third stripe} = 3 \]

5. **Continue This Pattern**: This pattern continues for the fourth, fifth, and sixth stripes. Each of these stripes will also have 3 choices, as they cannot be the same color as the stripe immediately before them.

\[ \text{Choices for the fourth stripe} = 3 \]
\[ \text{Choices for the fifth stripe} = 3 \]
\[ \text{Choices for the sixth stripe} = 3 \]

6. **Calculate the Total Number of Combinations**: To find the total number of ways to arrange the colors for the six stripes, we multiply the number of choices for each stripe together:

\[ \text{Total combinations} = 4 \times 3 \times 3 \times 3 \times 3 \times 3 \]

This can be simplified as:

\[ \text{Total combinations} = 4 \times 3^5 \]

7. **Calculate \(3^5\)**: Now we calculate \(3^5\):

\[ 3^5 = 243 \]

8. **Final Calculation**: Now we multiply 4 by 243:

\[ \text{Total combinations} = 4 \times 243 = 972 \]

Thus, the total number of ways to design the flag with the given conditions is **972**.