Question

Five persons P, Q, R, S and T are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and T cannot be seated at either end of the row. P should not be seated adjacent to S. R is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

When calculating seating arrangements with conditions, first fix the positions that are restricted (like R's position here) and then systematically consider the remaining conditions (like P and T not at the ends, and adjacency rules).

Answer 1

The Correct Option is B

We need to find the number of distinct seating arrangements given the following conditions:
- P and T cannot be seated at either end of the row.
- P should not be seated adjacent to S.
- R is to be seated at the second position from the left end of the row.

Step 1: Place R in the second position Since R is fixed in the second position, we now have the following positions for the remaining people: _ R _ _ _.

Step 2: Place P and T P and T cannot be seated at either end, so the only available positions for P and T are the 3rd and 4th positions. Therefore, we can place P and T in the 3rd and 4th positions in 2 ways (P in 3rd and T in 4th, or vice versa).

Step 3: Place S and Q Now, S and Q are left to be seated in the remaining two positions (the 1st and 5th positions). The condition is that P should not be adjacent to S, so S must be placed in the 5th position, and Q must be placed in the 1st position.

Step 4: Calculate the total arrangements The only possible arrangement is:
- P and T can be arranged in 2 ways in the 3rd and 4th positions.
- S and Q can be placed in the 1st and 5th positions in exactly 1 way (since S cannot sit next to P).
Thus, the total number of distinct seating arrangements is: \[ 2 + 1 = 3. \] Thus, the correct answer is Option (B).

Final Answer: (B) 3