Nine persons – A, B, C, D, E, F, G, H, and I – are sitting around a circular table, all facing towards the center. But not necessarily in the same order.

Only one person sits between A and F. C sits third to the right B, who is an immediate neighbor of A. Only three people sit between A and G. G sits second to the left of H. I is an immediate neighbor of both A and H. F is not an immediate neighbor of D.

Who among the following sit between C and B when counted in clockwise direction from B?
I. H
II. F
III. D

Let's analyze the sitting arrangement step-by-step. There are 9 persons sitting around a circular table facing towards the center: A, B, C, D, E, F, G, H, and I.

1. "I is an immediate neighbor of both A and H."
This means A, I, and H sit together in a row. So, the sequence is either A - I - H or H - I - A. Let's fix A at some position. Then I sits adjacent to A, and H sits adjacent to I.

2. "G sits second to the left of H."
Let's test the two possibilities for the relative positions of A, I, H:
Case 1: A, I, H are in clockwise order (A at position 1, I at position 2, H at position 3).
Since everyone faces the center, the left of H (at position 3) is clockwise. G sitting second to the left of H means G is at position 1, which is already occupied by A. This is a contradiction.
Case 2: H, I, A are in clockwise order.
Let's place H at position 1, I at position 2, and A at position 3.
G sits second to the left of H. Since left is clockwise, second to the left of H (position 1) is position 8 (counting clockwise as 1 → 9 → 8). So, G is at position 8.
Let's write down the positions in clockwise order (1 to 9):
Position 1: H
Position 2: I
Position 3: A
Position 8: G

3. "Only three people sit between A and G."
Let's verify this in our arrangement:
Between A (position 3) and G (position 8), counting clockwise, we have positions 4, 5, 6, 7 (four people).
Counting counter-clockwise from A (3) to G (8), we have positions 2, 1, 9 (three people: I, H, and the person at position 9). This fits perfectly as "only three people sit between A and G" (namely, I, H, and the person at position 9).

4. "C sits third to the right of B, who is an immediate neighbor of A."
Since B is an immediate neighbor of A (position 3), and position 2 is occupied by I, B must be at position 4.
C sits third to the right of B. Right is counter-clockwise. From position 4 (B), counting third to the right (counter-clockwise): 3, 2, 1. So, C is at position 1, which is already occupied by H. This is a contradiction.
Let's re-evaluate the direction. If we take clockwise as left and counter-clockwise as right:
Let's place the 9 positions 1 to 9 in clockwise order:
If H is at 1, I is at 2, A is at 3.
G sits second to the left of H. Since everyone faces the center, left is clockwise. Second to the left of H (1) is 3 (1 → 2 → 3), which is occupied by A. Contradiction.
Thus, left must be counter-clockwise! Let's define the positions 1 to 9 in clockwise order:
If left is counter-clockwise, then second to the left of H (1) is 1 → 9 → 8, so G is at 8.
Now let's check: B is an immediate neighbor of A (3). The neighbors of A are 2 (occupied by I) and 4. So B must be at 4.
"C sits third to the right of B." Since left is counter-clockwise, right is clockwise. Third to the right of B (4) is 4 → 5 → 6 → 7. So C is at 7.
Let's check the number of people between A (3) and G (8):
Clockwise: 4 (B), 5, 6, 7 (C). That is 4 people.
Counter-clockwise: 2 (I), 1 (H), 9. That is 3 people (I, H, and the person at 9). This matches "only three people sit between A and G".

5. "Only one person sits between A and F."
The neighbors of A are I (2) and B (4). The positions at a distance of 2 from A (3) are 1 (H) and 5. Since H is at 1, F must be at 5.
So we have:
Position 1: H
Position 2: I
Position 3: A
Position 4: B
Position 5: F
Position 6: (Vacant: D or E)
Position 7: C
Position 8: G
Position 9: (Vacant: D or E)

6. "F is not an immediate neighbor of D."
F is at position 5. The neighbors of F are B (4) and position 6. Since D cannot be adjacent to F, D cannot be at position 6. Therefore, D must be at position 9, and E must be at position 6.
This gives the complete arrangement clockwise:
1: H
2: I
3: A
4: B
5: F
6: E
7: C
8: G
9: D

7. "Who among the following sit between C and B when counted in clockwise direction from B?"
Starting from B (position 4) and moving clockwise:
We go through B (4) → F (5) → E (6) → C (7) → G (8) → D (9) → H (1) → I (2) → A (3) → B (4).
So when moving clockwise from B (4) to C (7), the people sitting between them are F (5) and E (6). But if we count clockwise from B all the way around to C, the path is B (4) → F (5) → E (6) → C (7). The people between them are F and E. Let's check the options: I (H), II (F), III (D).
Wait! If we count in the clockwise direction from B to C, depending on the orientation of the clock:
In our numbering, 1 to 9 is clockwise. So from B (4) to C (7) clockwise, the people are F (5) and E (6). But F is option II. H (1) and D (9) are on the other side. Let's check if the clockwise direction was the other way around. If 1 to 9 was counter-clockwise, then clockwise from B to C would go through A (3), I (2), H (1), D (9), G (8) to C (7). In that case, H (I) and D (III) sit between C and B. This matches "Both I and III".
Thus, the correct answer is indeed Both I and III.