Correct Option (D)
The problem requires identifying a statement that is definitively correct across all possible valid seating arrangements. A statement is "definitely correct" only if it holds true for every configuration that satisfies all given conditions. We must first derive the possible seating arrangements based on the provided constraints:
- Eight students (A, B, C, D, E, F, G, H) are seated around a circular table.
- Constraint 1: B and D are neither adjacent to C nor opposite to C.
- Constraint 2: A sits between E and D (implying a consecutive block of E-A-D or D-A-E).
- Constraint 3: F sits between B and H (implying a consecutive block of B-F-H or H-F-B).
Constraints 2 and 3 establish two blocks of three consecutive students: (E, A, D) and (B, F, H). This accounts for 6 students. The remaining two students are C and G. Given 8 seats in a circle, we can explore different relative placements of C and G, along with these two blocks.
Let's consider two distinct valid arrangements to test the options:
Arrangement 1: C and G are opposite.
Assume C is at seat 1. For C and G to be opposite, G must be at seat 5. The remaining six seats (2, 3, 4 and 6, 7, 8) are available for the two blocks of three. Let the block (E,A,D) occupy seats (2,3,4) and the block (B,F,H) occupy seats (6,7,8).
- To satisfy Constraint 1 (B and D not adjacent to C or opposite C):
- If D is at seat 4 (from E-A-D: 2-3-4), D is not adjacent to C (seat 1) nor opposite C (seat 5). This is valid.
- If B is at seat 6 (from B-F-H: 6-7-8), B is not adjacent to C (seat 1) nor opposite C (seat 5). This is valid.
A valid arrangement is: C, E, A, D, G, B, F, H (clockwise).
Arrangement 2: C and E are opposite.
Assume C is at seat 1. For C and E to be opposite, E must be at seat 5.
- From Constraint 2 (A sits between E and D): Since E is at seat 5, D must be at seat 4 or seat 6. If D is at seat 4, then A is at seat 3 (forming D-A-E: 4-3-5).
- The students placed so far are C(1), A(3), D(4), E(5). The remaining students are B, F, H, G. The remaining seats are 2, 6, 7, 8.
- From Constraint 3 (F sits between B and H): The block (B,F,H) requires three consecutive seats. Only seats (6,7,8) are available.
- The remaining student G must occupy seat 2.
- Now, check Constraint 1 (B and D not adjacent to C or opposite C):
- C is at seat 1. D is at seat 4. D is not adjacent to C (seats 2,8) nor opposite C (seat 5). This is valid.
- B must not be at seat 2, 8 (adjacent to C) or seat 5 (opposite C). Since (B,F,H) are at (6,7,8), B cannot be at seat 8 (as it would be adjacent to C). Therefore, B must be at seat 6 (forming B-F-H: 6-7-8). This is valid for B.
A valid arrangement is: C, G, A, D, E, B, F, H (clockwise).
We now evaluate the given options against these two distinct valid arrangements.
Incorrect Options:
Option (A): B sits in between A and G
- In Arrangement 1 (C, E, A, D, G, B, F, H), the sequence is A-D-G-B. B is not positioned between A and G.
- In Arrangement 2 (C, G, A, D, E, B, F, H), the sequence is G-A-D-E-B. B is not positioned between A and G.
This statement is false in both arrangements, thus it is not definitely correct.
Option (B): C sits opposite to G
- In Arrangement 1 (C, E, A, D, G, B, F, H), C is at seat 1 and G is at seat 5, indicating they are opposite. (True)
- However, in Arrangement 2 (C, G, A, D, E, B, F, H), C is at seat 1 and G is at seat 2, indicating they are adjacent. (False)
Since this statement is not true in all valid arrangements, it is not definitely correct.
Option (C): E sits opposite to F
- In Arrangement 1 (C, E, A, D, G, B, F, H), E is at seat 2 and F is at seat 7. They are not opposite (seat 2 is opposite seat 6).
- In Arrangement 2 (C, G, A, D, E, B, F, H), E is at seat 5 and F is at seat 7. They are not opposite (seat 5 is opposite seat 1).
This statement is false in both arrangements, thus it is not definitely correct.
As none of the options (A), (B), or (C) are definitively correct, the correct answer is (D) None of the above.