Number of ways in which out of `n` persons exactly `r` person can sit between two specified persons around a round a table is (where `n` is odd).

Options

A. Dependent on both `n` and `r`
B. Independent of both `n` and `r`
C. Dependent of `n` but Independent of `r`
D. Independent of `n` but dependent of `r`

Explanation

To solve the problem of determining the number of ways in which exactly `r` persons can sit between two specified persons around a round table, where `n` is odd, we can follow these steps:

### Step-by-Step Solution:

1. **Fix the Two Specified Persons**:
In a circular arrangement, we can fix the position of the two specified persons (let's call them A and B) to eliminate the circular symmetry. This means we can treat A and B as fixed points on the table.

**Hint**: Remember that in circular permutations, fixing one position helps simplify the arrangement.

2. **Identify the Remaining Persons**:
After fixing A and B, we have `n - 2` persons left (since A and B are already seated).

**Hint**: Count how many persons are left after placing the specified ones.

3. **Choose `r` Persons to Sit Between A and B**:
We need to choose `r` persons from the remaining `n - 2` persons to sit between A and B. The number of ways to choose `r` persons from `n - 2` is given by the combination formula:
\[ \binom{n-2}{r} \]

**Hint**: Use the combination formula to select the required number of persons.

4. **Arrange the Chosen `r` Persons**:
The `r` persons that we have chosen can be arranged in `r!` different ways between A and B.

**Hint**: Remember that the arrangement of chosen persons matters, so use factorial.

5. **Arrange the Remaining Persons**:
After seating the `r` persons, there are `n - 2 - r` persons left. These remaining persons can be seated in the remaining seats around the table. The number of ways to arrange these remaining persons is:
\[ (n - 2 - r)! \]

**Hint**: Factorial is used here for arranging the remaining persons.

6. **Combine All Parts**:
The total number of arrangements can be calculated by multiplying the number of ways to choose the `r` persons, the arrangements of those `r` persons, and the arrangements of the remaining persons:
\[ \text{Total Ways} = \binom{n-2}{r} \cdot r! \cdot (n - 2 - r)! \]

**Hint**: Combine the results from previous steps to get the final answer.

7. **Simplify the Expression**:
We can simplify the expression:
\[ \text{Total Ways} = \frac{(n-2)!}{r! \cdot (n-2-r)!} \cdot r! \cdot (n - 2 - r)! = (n - 2)! \]

**Hint**: Notice how the factorials cancel out to simplify the expression.

### Final Answer:
Thus, the number of ways in which out of `n` persons exactly `r` persons can sit between two specified persons around a round table is:
\[ (n - 2)! \]