Eight people sit in a minibus: 4 on the sunny side and 4 on the shady side. If 2 people want to sit on opposite sides to each other, another 2 people want to sit on the shady side, in how many ways can this be done?

To solve this problem, let's break it down into manageable steps and use combinatorial methods.

1. **Label the seats and people**: Let's label the seats on the sunny side as \( S_1, S_2, S_3, S_4 \) and the seats on the shady side as \( H_1, H_2, H_3, H_4 \). Let's label the people as \( P_1, P_2, P_3, P_4, P_5, P_6, P_7, P_8 \).

2. **Identify constraints**:
   - \( P_1 \) and \( P_2 \) want to sit on opposite sides.
   - \( P_3 \) and \( P_4 \) want to sit on the shady side.

3. **Arrange \( P_1 \) and \( P_2 \)**:
   - \( P_1 \) can sit on any of the 4 sunny side seats.
   - For each choice of \( P_1 \), \( P_2 \) must sit on one of the 4 shady side seats.
   - This gives \( 4 \times 4 = 16 \) ways to place \( P_1 \) and \( P_2 \).

4. **Arrange \( P_3 \) and \( P_4 \)**:
   - They need to sit on the shady side.
   - There are 4 shady side seats, and \( P_2 \) already occupies one of them.
   - This leaves 3 seats for \( P_3 \) and \( P_4 \), and they can be arranged in \(\binom{3}{2} \times 2! = 3 \times 2 = 6 \) ways.

5. **Arrange the remaining 4 people (\( P_5, P_6, P_7, P_8 \))**:
   - After placing \( P_1 \), \( P_2 \), \( P_3 \), and \( P_4 \), we have 2 sunny side seats and 1 shady side seat left (3 seats remaining).
   - We can choose any 2 out of these 4 people to sit in the remaining sunny side seats. This can be done in \(\binom{4}{2} = 6 \) ways.
   - The remaining 2 people will sit on the 2 available seats (one shady and one sunny). These two remaining people can be arranged in \(2!\) ways.

6. **Combine all choices**:
   - Number of ways to place \( P_1 \) and \( P_2 \) = 16.
   - Number of ways to place \( P_3 \) and \( P_4 \) = 6.
   - Number of ways to place the remaining 4 people = 6 \times 2 = 12.

Hence, the total number of ways is:
\[ 16 \times 6 \times 12 = 1152 \]

So, the number of ways to arrange the people satisfying all conditions is \( \boxed{1152} \).

Would you like more details or have any questions? Here are some additional questions you might find interesting:

1. How would the arrangement change if more people had seating preferences?
2. What if the minibus had a different number of seats on each side?
3. How can we generalize this solution for any number of people?
4. What are other combinatorial constraints that can be introduced in this problem?
5. How would the solution change if specific people wanted to sit together?

**Tip**: When solving combinatorial problems, break the problem into smaller parts and handle constraints step-by-step to simplify the process.

Explore how to solve a complex seating arrangement problem involving preferences and constraints. Use combinatorial methods to find the number of ways to arrange people in a minibus with specific seating preferences.

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