A committee of 11 members sits at a round table. In how many ways can they be seated if the 'President' and the 'Secretary' choose to sit together?

Answer

To solve the problem of seating a committee of 11 members at a round table, where the 'President' and 'Secretary' choose to sit together, we can follow these steps:

### Step-by-Step Solution:

1. **Consider the President and Secretary as One Unit**:
Since the President and Secretary want to sit together, we can treat them as a single unit or block. This means instead of 11 individual members, we now have 10 units to arrange (the block of President and Secretary plus the other 9 members).

2. **Calculate the Arrangements of the Units**:
When arranging units in a circle, the formula we use is (n - 1)!, where n is the number of units. Here, we have 10 units (the block + 9 other members), so we calculate:
\[ (10 - 1)! = 9! \]

3. **Arrange the President and Secretary Within Their Block**:
The President and Secretary can switch places within their block. There are 2 ways to arrange them (President first or Secretary first). Thus, we multiply the arrangements of the units by the arrangements within the block:
\[ 2! = 2 \]

4. **Combine the Results**:
Now, we multiply the number of arrangements of the units by the arrangements within the block:
\[ \text{Total arrangements} = 9! \times 2! \]
Simplifying this gives:
\[ = 9! \times 2 \]

5. **Final Calculation**:
The final answer can be expressed as:
\[ 9! \times 2 = 362880 \times 2 = 725760 \]

### Final Answer:
The total number of ways the committee can be seated at the round table, with the President and Secretary sitting together, is **725760**.