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. The number of distinct seating arrangements possible is:

The correct option is 16.

To find the number of distinct seating arrangements, we can break the problem down into cases based on the positions of Q and R, since they must be separated by exactly one person. Let the five seats be numbered 1, 2, 3, 4, and 5 from left to right. There are three possible pairs of positions that Q and R can occupy to remain separated by exactly one seat:
1. Seats 1 and 3
2. Seats 2 and 4
3. Seats 3 and 5

Let us analyze each case in detail, keeping in mind that S cannot sit adjacent to Q.

Case 1: Q and R occupy seats 1 and 3

There are two subcases depending on the order of Q and R:

Subcase 1.1: Q is at seat 1 and R is at seat 3
The empty seats are 2, 4, and 5. Since Q is at seat 1, S cannot sit at the adjacent seat, which is seat 2. Therefore, S must sit at either seat 4 or seat 5 (2 options).
Once S is seated, the remaining two people (P and T) can be seated in the remaining two empty seats in:
2! = 2 ways.
So, the number of arrangements for this subcase is:
2 × 2 = 4 arrangements.

Subcase 1.2: R is at seat 1 and Q is at seat 3
The empty seats are 2, 4, and 5. Since Q is at seat 3, S cannot sit at the adjacent seats, which are seat 2 and seat 4. Therefore, S must sit at seat 5 (1 option).
The remaining two seats (2 and 4) can be occupied by P and T in:
2! = 2 ways.
So, the number of arrangements for this subcase is:
1 × 2 = 2 arrangements.

Thus, the total arrangements for Case 1 is:
4 + 2 = 6 arrangements.

Case 2: Q and R occupy seats 2 and 4

There are two subcases depending on the order of Q and R:

Subcase 2.1: Q is at seat 2 and R is at seat 4
The empty seats are 1, 3, and 5. Since Q is at seat 2, S cannot sit adjacent to Q (seats 1 and 3 are adjacent to seat 2). Thus, S must sit at seat 5 (1 option).
The remaining two seats (1 and 3) can be occupied by P and T in:
2! = 2 ways.
So, the number of arrangements for this subcase is:
1 × 2 = 2 arrangements.

Subcase 2.2: R is at seat 2 and Q is at seat 4
The empty seats are 1, 3, and 5. Since Q is at seat 4, S cannot sit adjacent to Q (seats 3 and 5 are adjacent to seat 4). Thus, S must sit at seat 1 (1 option).
The remaining two seats (3 and 5) can be occupied by P and T in:
2! = 2 ways.
So, the number of arrangements for this subcase is:
1 × 2 = 2 arrangements.

Thus, the total arrangements for Case 2 is:
2 + 2 = 4 arrangements.

Case 3: Q and R occupy seats 3 and 5

This case is symmetrical to Case 1 due to the mirror layout of the seats.

Subcase 3.1: Q is at seat 5 and R is at seat 3
Since Q is at seat 5, S cannot sit at the adjacent seat (seat 4). Thus, S can sit at either seat 1 or seat 2 (2 options). The other two seats are filled by P and T in 2! = 2 ways.
Arrangements = 2 × 2 = 4 arrangements.

Subcase 3.2: R is at seat 5 and Q is at seat 3
Since Q is at seat 3, S cannot sit at seat 2 or seat 4. Thus, S must sit at seat 1 (1 option). The other two seats are filled by P and T in 2! = 2 ways.
Arrangements = 1 × 2 = 2 arrangements.

Thus, the total arrangements for Case 3 is:
4 + 2 = 6 arrangements.

Total Distinct Arrangements

Summing up the successful seating arrangements from all three cases, we get:
Total arrangements = Case 1 + Case 2 + Case 3
Total arrangements = 6 + 4 + 6 = 16