Question

Four persons P, Q, R and S are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and R cannot sit adjacent to each other. S should be seated to the right of Q. The number of distinct seating arrangements possible is:

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

When calculating seating arrangements with restrictions, first calculate the total number of arrangements and then subtract the invalid cases that violate the given conditions.

Answer 1

The Correct Option is C

We need to calculate the number of valid seating arrangements for four persons, P, Q, R, and S, where the following conditions apply: - P and R cannot sit adjacent to each other. - S must be seated to the right of Q.

Step 1: Calculate the total number of seating arrangements without restrictions.
The total number of ways to arrange 4 persons in a row is \( 4! = 24 \).

Step 2: Subtract the number of arrangements where P and R sit adjacent to each other.
Treat P and R as a single "block," so we have 3 "objects" to arrange (the PR block, Q, and S). The number of ways to arrange these 3 objects is \( 3! = 6 \). Since P and R can be arranged within the block in 2 ways (P first or R first), the total number of arrangements where P and R sit adjacent is \( 6 \times 2 = 12 \).

Step 3: Apply the condition that S must be seated to the right of Q.
For the remaining valid arrangements, we need to consider only those where S is seated to the right of Q. Out of the total 12 arrangements, half will satisfy this condition, so there are \( \frac{12}{2} = 6 \) valid seating arrangements.

Final Answer: 6.