6th year students at Escola Flor de Liz will participate in a competition in which they will have to create a logo to represent the school's football team. The students chose one format consisting of 6 horizontal and adjacent bands. To paint the logo they have 4 different colors available: red, blue, green and yellow. How many ways are there to paint the team logo so that the adjacent stripes have different colors?

To solve the problem of determining how many ways there are to paint the team's logo such that adjacent stripes have different colors, we can approach it as follows:

1. **Understand the Problem Setup:**
   - The logo consists of 6 horizontal and adjacent bands.
   - There are 4 colors available: red, blue, green, and yellow.
   - No two adjacent bands should have the same color.

2. **Counting the Possible Colorings:**
   - For the first band, there are 4 possible colors to choose from.
   - For each subsequent band, you can choose any of the remaining 3 colors (since the only restriction is that it must differ from the color of the previous band).

Thus, the total number of ways to paint the logo can be calculated as:

\[
4 \times 3 \times 3 \times 3 \times 3 \times 3 = 4 \times 3^5
\]

3. **Calculate the Number:**

\[
3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243
\]

\[
4 \times 243 = 972
\]

So, there are **972** different ways to paint the logo so that adjacent stripes have different colors.

Would you like more details on how this was calculated, or do you have any questions?

Here are some related questions you might find interesting:
1. What if there were only 3 colors available?
2. How many ways would there be to paint 5 stripes instead of 6?
3. What happens if the restriction on adjacent colors is removed?
4. How many different color patterns are there if the bands can have the same color?
5. If the logo had vertical stripes instead of horizontal ones, would the number of ways change?
6. How does the number of bands affect the number of colorings?
7. If one color must be used exactly twice, how would that change the calculation?
8. Can we generalize this method for any number of bands and colors?

**Tip:** When solving problems involving restrictions, it's often helpful to calculate the number of possibilities for each step, taking into account the restrictions as you proceed.

Learn how to calculate the number of ways to paint a logo with 6 adjacent horizontal bands using 4 different colors, ensuring no two adjacent bands have the same color. This combinatorics problem involves permutations and combinations, suitable for high school students.

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