A cabinet of ministers consists of 11 ministers, one minister being the chief minister. A meeting is to be held in a room having a round table and 11 chairs round it, one of them being meant for the chairman. The number of ways in which the ministers can take their chairs, the chief minister occupying the chairman’s place,

Options

A. `(1)/(2)(10!)`
B. `9!`
C. `10!`
D. `(9!)/(2)`

Answer

To solve the problem, we need to determine the number of ways the ministers can take their seats around a round table, given that the chief minister occupies a designated chair meant for the chairman.

### Step-by-Step Solution:

1. **Identify the Fixed Position**:
Since the chief minister occupies the chairman's chair, this position is fixed. This means we do not need to consider arrangements for this chair.

2. **Count Remaining Ministers and Chairs**:
After the chief minister is seated, we have 10 remaining ministers and 10 remaining chairs to arrange them in.

3. **Understanding Arrangements**:
The arrangement of the remaining ministers can be treated as a linear arrangement since the chairman's position is fixed. Thus, we need to arrange 10 ministers in 10 chairs.

4. **Calculate the Number of Arrangements**:
The number of ways to arrange 10 ministers in 10 chairs is given by the factorial of the number of ministers. This can be expressed mathematically as:
\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

5. **Final Answer**:
Therefore, the total number of ways the ministers can take their seats, with the chief minister in the chairman's position, is \(10!\).

### Conclusion:
The answer to the problem is \(10!\).

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