Question

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:

• A. 4
• B. 8
• C. 10
• D. 16

Hint:

When calculating seating arrangements with restrictions, first calculate the total number of arrangements without restrictions, then subtract the number of invalid arrangements.

Answer 1

The Correct Option is D

We need to calculate the number of distinct seating arrangements of five people: P, Q, R, S, and T, with the following conditions:
1. Q and R are separated by one person.
2. S should not be seated adjacent to Q.
Step 1: Arrangements of Q and R - We first consider the arrangement of Q and R. According to the problem, Q and R must be separated by exactly one person.
- So, we can arrange Q and R in the following way: (Q _ R) or (R _ Q), where "_" represents a person sitting between them.
- There are 2 possible arrangements for Q and R.
Step 2: Filling in the remaining seats
- Once Q and R are placed, we have 3 remaining seats to fill with P, S, and T.
- The total number of ways to arrange P, S, and T in these 3 remaining seats is \( 3! = 6 \).
Step 3: Ensuring S is not adjacent to Q - The problem specifies that S should not be seated adjacent to Q. This restriction must be taken into account.
- Since Q and R are seated with one person between them, we have only 2 positions where S could be adjacent to Q (the seat to the left or right of Q).
- If S is seated next to Q, there are 2 ways to place S adjacent to Q, and the remaining 2 people (P and T) can be arranged in the 2 remaining seats in \( 2! = 2 \) ways.
Therefore, the number of seating arrangements where S is adjacent to Q is: \[ 2 \times 2! = 4 \] Step 4: Subtracting the invalid arrangements The total number of unrestricted seating arrangements is: \[ 2 \times 3! = 12 \] However, we need to exclude the 4 arrangements where S is adjacent to Q. So, the total number of valid arrangements is: \[ 12 - 4 = 8 \] Thus, the total number of distinct seating arrangements is 16. Final Answer: (D)