If Kathryn is designing a flag similar to the French flag, where no 2 adjacent stripes must be colored the same, and there are 7 available colors, we can consider the following:
1. The first stripe can be any of the 7 colors.
2. The second stripe cannot be the same color as the first one, so it has 6 choices.
3. The third stripe must be different from the second one, so it has 6 choices.
4. This pattern continues for the remaining stripes.
The total number of possible flags can be calculated by multiplying the number of choices for each stripe: [tex]\[ 7 \times 6 \times 6 \times \ldots \][/tex]
If there are [tex]\( n \)[/tex] stripes, the total number of possible flags is [tex]\( 7 \times 6^{(n-1)} \)[/tex]
For example, Kathryn wants to design a flag with 3 stripes, the total number of possible flags is [tex]\( 7 \times 6^2 = 252 \)[/tex]
Therefore, the number of possible flags depends on the number of stripes Kathryn wants in her design.
