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:
2
4
3
5
The correct answer is 3.
Step-by-Step Explanation:
Let us represent the five seating positions from left to right as five slots:
Slot 1, Slot 2, Slot 3, Slot 4, Slot 5
1. Position of R:
We are given that R is seated at the second position from the left end of the row (Slot 2).
So, the arrangement looks like:
_ R _ _ _
2. Positions of P and T:
We are given that P and T cannot be seated at either end of the row (Slot 1 and Slot 5).
Since R is already at Slot 2, the only remaining middle slots where P and T can sit are Slot 3 and Slot 4.
Therefore, P and T must occupy Slot 3 and Slot 4 in some order. This gives two main configurations:
• Configuration 1: P is at Slot 3 and T is at Slot 4 (_ R P T _)
• Configuration 2: T is at Slot 3 and P is at Slot 4 (_ R T P _)
3. Positions of Q and S:
For both configurations, the remaining two individuals, Q and S, must occupy the ends of the row (Slot 1 and Slot 5) in some order.
Let's analyze the validity of the arrangements under each configuration:
Case 1: P is at Slot 3 and T is at Slot 4 (_ R P T _)
• Sub-case 1a: Q is at Slot 1 and S is at Slot 5.
Arrangement: Q R P T S
Here, P (Slot 3) and S (Slot 5) are not adjacent. Thus, this arrangement is valid.
• Sub-case 1b: S is at Slot 1 and Q is at Slot 5.
Arrangement: S R P T Q
Here, P (Slot 3) and S (Slot 1) are not adjacent. Thus, this arrangement is valid.
Case 2: T is at Slot 3 and P is at Slot 4 (_ R T P _)
• Sub-case 2a: Q is at Slot 1 and S is at Slot 5.
Arrangement: Q R T P S
Here, P (Slot 4) and S (Slot 5) are adjacent. Since P and S cannot be adjacent, this arrangement is invalid.
• Sub-case 2b: S is at Slot 1 and Q is at Slot 5.
Arrangement: S R T P Q
Here, P (Slot 4) and S (Slot 1) are not adjacent. Thus, this arrangement is valid.
Analysis of the Attached Diagram:
The attached image details the drafting rules:
1. P and T cannot be at the ends.
2. P and S cannot be adjacent (indicated by a line with a cross between P and S).
3. R is placed at the 2nd slot from the left.
The image lists the resulting possibilities as:
• Q R P T S (Valid)
• S R P T Q (Valid)
• Q R T P S (Note: This is a slight typo in the image's list; it should be S R T P Q to keep P and S non-adjacent, but the total count remains correct).
Thus, the total number of distinct valid seating arrangements is exactly 3.
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