If 11 members of a committee sit at a round table so that the president and secretary always sit together, then the number of arrangements, is

Options

A. `9!xx2`
B. `10!`
C. `10!xx2`
D. `11!`

Answer

To solve the problem of arranging 11 members of a committee at a round table with the condition that the president and secretary always sit together, we can follow these steps:

### Step-by-Step Solution:

1. **Treat the President and Secretary as a Single Unit**:
Since the president and secretary must sit together, we can consider them as one single unit or block. This means instead of 11 individual members, we now have 10 units to arrange: the block (president + secretary) and the other 9 members.

2. **Calculate the Arrangements in a Circular Table**:
The formula for arranging \( n \) people in a circular arrangement is \( (n-1)! \). Here, since we have 10 units (the block plus 9 other members), the number of ways to arrange these 10 units in a circle is:
\[ (10 - 1)! = 9! \]

3. **Account for the Internal Arrangement of the Block**:
Within the block, the president and secretary can switch places. Therefore, for each arrangement of the 10 units, there are 2 ways to arrange the president and secretary within their block. Thus, we multiply the arrangements by 2:
\[ 2! \]

4. **Combine the Two Calculations**:
Now, we multiply the number of arrangements of the 10 units by the arrangements of the president and secretary:
\[ \text{Total arrangements} = 9! \times 2! \]

5. **Calculate the Final Result**:
Now we can compute the values:
\[ 9! = 362880 \quad \text{and} \quad 2! = 2 \]
Therefore:
\[ \text{Total arrangements} = 362880 \times 2 = 725760 \]

### Final Answer:
The total number of arrangements of the committee members at the round table, with the president and secretary sitting together, is **725760**.