In how many ways can the eight directors, the vice- chairman and the chairman of a firm be seated at a round- table, if the chairman has to sit between the vice-chairman and the director?

Options

A. `9! xx 2`
B. `2 xx 8!`
C. `2 xx 7!`
D. None of these

Answer

To solve the problem of how many ways the eight directors, the vice-chairman, and the chairman can be seated at a round table with the condition that the chairman must sit between the vice-chairman and one of the directors, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of People**:
We have 8 directors, 1 vice-chairman, and 1 chairman. Therefore, the total number of people is:
\[ 8 + 1 + 1 = 10 \]

2. **Fix the Arrangement for the Chairman and Vice-Chairman**:
Since the chairman must sit between the vice-chairman and one of the directors, we can treat the chairman and vice-chairman as a block or a set. This means we will consider the arrangement of these three individuals (Vice-Chairman, Chairman, Director) as a single unit.

3. **Count the Arrangement of the Block**:
The arrangement of the block (Vice-Chairman, Chairman, Director) can be done in two ways:
- Vice-Chairman, Chairman, Director
- Director, Chairman, Vice-Chairman

Thus, there are 2 arrangements for this block.

4. **Calculate the Remaining Directors**:
After fixing the block, we have 7 remaining directors to arrange along with the block. Therefore, we have a total of 8 units to arrange (the block + 7 directors).

5. **Arrange the Units in a Round Table**:
For arranging \( n \) people in a round table, the number of arrangements is given by \( (n-1)! \). Here, we have 8 units (the block and the 7 directors), so the number of arrangements is:
\[ (8-1)! = 7! \]

6. **Combine the Arrangements**:
Now, we multiply the arrangements of the block by the arrangements of the units:
\[ \text{Total arrangements} = 2 \times 7! \]

### Final Calculation:
Thus, the total number of ways to arrange the directors, vice-chairman, and chairman at the round table is:
\[ 2 \times 7! = 2 \times 5040 = 10080 \]

### Conclusion:
The total number of ways the eight directors, the vice-chairman, and the chairman can be seated at a round table, with the chairman sitting between the vice-chairman and a director, is **10080**.