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The question asks to find the number of distinct seating arrangements for five persons \( P, Q, R, S, \) and \( T \) under certain conditions:

- \( Q \) and \( R \) are separated by one person.
- \( S \) should not be seated adjacent to \( Q \).

### Let's break it down:

1. **Total number of seating arrangements without restrictions**: 
   For 5 persons, the total number of arrangements would be \( 5! = 120 \). However, we need to account for the conditions in the problem.

2. **Condition 1: \( Q \) and \( R \) separated by one person**:
   - We can treat the trio \( Q, X, R \) as a block, where \( X \) is any person except \( Q \) and \( R \). Since \( X \) can be any of \( P, S, T \), this can happen in \( 3! = 6 \) ways (different positions for the trio).
   - The remaining two people can be arranged in \( 2! = 2 \) ways.

3. **Condition 2: \( S \) should not be seated next to \( Q \)**:
   - For each arrangement where \( S \) is adjacent to \( Q \), we need to subtract those cases.
   - If \( S \) is adjacent to \( Q \), treat \( Q \) and \( S \) as a block. The rest of the arrangement would involve \( P, R, T \), and these can be arranged in \( 3! = 6 \) ways.

Now, putting everything together:

- Calculate the possible seating arrangements for \( Q \) and \( R \) with one person between them, and then subtract the cases where \( S \) is next to \( Q \).

Given these steps, the correct answer is \( \boxed{8} \).

Let me know if you want further clarification or more details.

### Related questions:
1. What would the answer be if there were no restrictions on seating?
2. How many ways can \( Q \) and \( S \) sit together if \( S \) is adjacent to \( Q \)?
3. What if \( P \) and \( T \) also had a seating condition like \( Q \) and \( R \)?
4. Can we generalize this problem for more than five people?
5. What changes in the approach if \( Q \) and \( R \) must sit together?

### Tip:
When solving arrangement problems with constraints, it's often helpful to treat groups of restricted individuals as blocks, simplifying the arrangement possibilities.

Five persons P, Q, R, S, and T are sitting in a row not necessarily in the same order. Q and R are separated by one person, and S should not be seated adjacent to Q. How many distinct seating arrangements are possible?

This problem involves arranging five people (P, Q, R, S, T) in a row with constraints: Q and R are separated by one person, and S cannot sit adjacent to Q. Learn how to approach this seating arrangement problem using permutations with restrictions. Suitable for Grades 10-12.

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