Five people A, B, C, D and E are seated about a round table. Every chair is spaced equidistant from adjacent chairs.
(i) C is seated next to A
(ii) A is seated two seats from D
(iii) B is not seated next to A
On the basis of above information, which of the following must be true ?
1. D is seated next to B
2. E is sealed next to A .
3. D and C are separated by two seats
Select the correct answer using the code given below:

The correct option is 1 and 2 only.

Let us analyze the seating arrangement step-by-step to understand why this option is correct. Since there are five people (A, B, C, D, and E) seated around a round table with equidistant chairs, we can represent their positions using numbers 1 to 5 in a clockwise order, where seat 1 is adjacent to seats 2 and 5.

Without loss of generality, let us place A at seat 1. Now, we analyze the constraints step-by-step:

Step 1: Place C based on condition (i)
Condition (i) states that C is seated next to A. Since A is at seat 1, the seats next to A are seat 2 and seat 5. This gives us two main cases:
• Case 1: C is at seat 2.
• Case 2: C is at seat 5.

Step 2: Place D based on condition (ii)
Condition (ii) states that A is seated two seats from D. From seat 1, moving two seats (either clockwise or counter-clockwise) leads to seat 3 or seat 4. Therefore, D must be seated at either seat 3 or seat 4.

Step 3: Analyze Case 1 (C is at seat 2)
We have two sub-cases for the position of D:
• Sub-case 1a: D is at seat 3.
The remaining empty seats are seat 4 and seat 5, which must be occupied by B and E. According to condition (iii), B is not seated next to A. Since seat 5 is adjacent to A (seat 1), B cannot be at seat 5. Therefore, B must be at seat 4, leaving seat 5 for E. The resulting arrangement is:
A at 1, C at 2, D at 3, B at 4, and E at 5.
• Sub-case 1b: D is at seat 4.
The remaining empty seats are seat 3 and seat 5. Since B cannot be next to A (seat 5), B must be at seat 3, leaving seat 5 for E. The resulting arrangement is:
A at 1, C at 2, B at 3, D at 4, and E at 5.

Step 4: Analyze Case 2 (C is at seat 5)
Similarly, we have two sub-cases for the position of D:
• Sub-case 2a: D is at seat 3.
The remaining empty seats are seat 2 and seat 4. Since B cannot be next to A (seat 2), B must be at seat 4, leaving seat 2 for E. The resulting arrangement is:
A at 1, E at 2, D at 3, B at 4, and C at 5.
• Sub-case 2b: D is at seat 4.
The remaining empty seats are seat 2 and seat 3. Since B cannot be next to A (seat 2), B must be at seat 3, leaving seat 2 for E. The resulting arrangement is:
A at 1, E at 2, B at 3, D at 4, and C at 5.

Step 5: Evaluate the statements against all possible arrangements
Let us check the three statements for all four valid seating arrangements:
1. D is seated next to B: In all four arrangements, D and B are placed at adjacent seats (seats 3 and 4). Thus, statement 1 is must be true.
2. E is seated next to A: In all four arrangements, E is placed at either seat 2 or seat 5, both of which are adjacent to A (seat 1). Thus, statement 2 is must be true.
3. D and C are separated by two seats: In our arrangements, D and C are either adjacent (separated by 0 seats) or separated by 1 seat. Thus, statement 3 is false.

Therefore, only statements 1 and 2 must be true.