Eight students A, B, C, D, E, F, G and H sit around a circular table, equidistant from each other, facing the centre of the table, not necessarily in the same order. Band D sit neither adjacent to C nor opposite to C . Asits in between E and D , and F sits in between B and H . Which one of the following is definitely correct?

To determine which statement is definitely correct, we need to find all possible valid seating arrangements based on the given conditions and then check each option against them.

Let's denote the 8 positions around the circular table from 1 to 8 in a clockwise direction. We can fix C's position without loss of generality, say at position 1.

The conditions are:

1. B and D sit neither adjacent to C nor opposite to C.
   • If C is at position 1, B and D cannot be at positions 2, 8 (adjacent) or 5 (opposite).
   • So, B and D must be from positions {3, 4, 6, 7}.
2. A sits in between E and D. This means E, A, D are consecutive (E-A-D or D-A-E).
3. F sits in between B and H. This means B, F, H are consecutive (B-F-H or H-F-B).

Let's explore possible arrangements:

Arrangement 1:

• Let C be at position 1.
• Consider D at position 4. From condition 1, this is allowed (not 2, 8, or 5).
• From condition 2 (E-A-D), if D is at 4, then A is at 3 and E is at 2.
  • Partial arrangement: C(1)-E(2)-A(3)-D(4)----
• Remaining positions: 5, 6, 7, 8. Remaining students: B, F, H, G.
• From condition 1, B cannot be at 5 (opposite C) or 8 (adjacent to C). So B must be at 6 or 7.
• If B is at 6: From condition 3 (B-F-H), F must be at 7 and H at 8.
• The remaining student G must be at position 5.
• This gives Arrangement 1: C(1)-E(2)-A(3)-D(4)-G(5)-B(6)-F(7)-H(8).
• Let's verify this arrangement:
  • 1. B(6) and D(4) are not adjacent or opposite to C(1). (Correct)
  • 2. A(3) is between E(2) and D(4). (Correct)
  • 3. F(7) is between B(6) and H(8). (Correct)
  • Arrangement 1 is valid.

Arrangement 2:

• Let C be at position 1.
• Consider D at position 6. From condition 1, this is allowed.
• From condition 2 (E-A-D), if D is at 6, then A is at 5 and E is at 4.
  • Partial arrangement: C(1)--E(4)-A(5)-D(6)--
• Remaining positions: 2, 3, 7, 8. Remaining students: B, F, H, G.
• From condition 1, B cannot be at 2, 8 (adjacent to C) or 5 (opposite C, but 5 is A). So B must be at 3 or 7.
• If B is at 3: From condition 3 (H-F-B), F must be at 2 and H at 8.
• The remaining student G must be at position 7.
• This gives Arrangement 2: C(1)-F(2)-B(3)-E(4)-A(5)-D(6)-G(7)-H(8).
• Let's verify this arrangement:
  • 1. B(3) and D(6) are not adjacent or opposite to C(1). (Correct)
  • 2. A(5) is between E(4) and D(6). (Correct)
  • 3. F(2) is between H(8) and B(3). (Correct)
  • Arrangement 2 is valid.

Now let's evaluate each option using these two valid arrangements. For an option to be "definitely correct," it must be true in ALL possible valid arrangements.

A) B sits in between A and G * In Arrangement 1 (C-E-A-D-G-B-F-H): A is at 3, G is at 5, B is at 6. B is not between A and G. (False) * In Arrangement 2 (C-F-B-E-A-D-G-H): A is at 5, G is at 7, B is at 3. B is not between A and G. (False) * Since this statement is false in both arrangements, it is not definitely correct.

B) C sits opposite to G * In Arrangement 1 (C-E-A-D-G-B-F-H): C is at 1, G is at 5. They are opposite. (True) * In Arrangement 2 (C-F-B-E-A-D-G-H): C is at 1, G is at 7. They are not opposite (1 is opposite 5). (False) * Since this statement is false in at least one arrangement, it is not definitely correct.

C) E sits opposite to F * In Arrangement 1 (C-E-A-D-G-B-F-H): E is at 2, F is at 7. They are not opposite (2 is opposite 6). (False) * In Arrangement 2 (C-F-B-E-A-D-G-H): E is at 4, F is at 2. They are not opposite (4 is opposite 8). (False) * Since this statement is false in both arrangements, it is not definitely correct.

Since none of the options A, B, or C are definitely correct, the answer must be D.

The final answer is $\boxed{D}$