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 seated next to A 3. D and C are separated by two seats Select the correct answer using the code given below:

The key to solving this problem lies in interpreting the phrase "A is seated two seats from D". In the context of a circular table with equidistant chairs, "two seats from" typically means there are two people between them.

Let's place A at a fixed position (e.g., position 1). There are 5 chairs: 1-2-3-4-5 (clockwise).

From condition (ii) "A is seated two seats from D": This means there are two people between A and D. If A is at position 1, D must be at position 4 (with positions 2 and 3 between them). Arrangement so far: A _ _ D _ (1 _ _ 4 _)

From condition (i) "C is seated next to A": C can be at position 2 or position 5.

Case 1: C is at position 2. Arrangement: A C _ D _ (1 2 _ 4 _) Remaining seats: 3, 5. Remaining people: B, E. From condition (iii) "B is not seated next to A": A is at 1, so B cannot be at 2 or 5. Since C is at 2, B cannot be at 2. So B cannot be at 5. Therefore, B must be at position 3. This leaves E for position 5. Arrangement 1: A C B D E (clockwise: A-C-B-D-E) Let's verify this arrangement: (i) C next to A (A-C): Yes (ii) A two seats from D (A-C-B-D): Yes, C and B are between A and D. (iii) B not next to A (A-C-B): Yes, C is between A and B.

Case 2: C is at position 5. Arrangement: A _ _ D C (1 _ _ 4 5) Remaining seats: 2, 3. Remaining people: B, E. From condition (iii) "B is not seated next to A": A is at 1, so B cannot be at 2 or 5. Since C is at 5, B cannot be at 5. So B cannot be at 2. Therefore, B must be at position 3. This leaves E for position 2. Arrangement 2: A E B D C (clockwise: A-E-B-D-C) Let's verify this arrangement: (i) C next to A (C-A): Yes (ii) A two seats from D (A-E-B-D): Yes, E and B are between A and D. (iii) B not next to A (A-E-B): Yes, E is between A and B.

Both Arrangement 1 (A C B D E) and Arrangement 2 (A E B D C) are valid. Now let's check the given statements:

1. D is seated next to B

   • In Arrangement 1 (A C B D E): B is at 3, D is at 4. Yes, they are next to each other.
   • In Arrangement 2 (A E B D C): B is at 3, D is at 4. Yes, they are next to each other. Since this is true in both valid arrangements, statement 1 MUST be true.

2. E is seated next to A

   • In Arrangement 1 (A C B D E): E is at 5, A is at 1. Yes, they are next to each other.
   • In Arrangement 2 (A E B D C): E is at 2, A is at 1. Yes, they are next to each other. Since this is true in both valid arrangements, statement 2 MUST be true.

3. D and C are separated by two seats

   • In Arrangement 1 (A C B D E): C is at 2, D is at 4. B is between them (1 person). So they are separated by one seat, not two. This statement is false for Arrangement 1.
   • In Arrangement 2 (A E B D C): C is at 5, D is at 4. They are next to each other (0 people). This statement is false for Arrangement 2. Since this is false in both valid arrangements, statement 3 is NOT necessarily true.

Therefore, only statements 1 and 2 must be true.

The final answer is B) 1 and 2 only.