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.
Consider the President and the Secretary as $1$ person.
Then there are $10$ people to be seated at a round table.
This can be done in $\left(10 - 1\right)! = 9!$ ways.
Amongst themselves, the President and the Secretary can be seated in $2$ ways.
$\therefore \;$ Number of ways a committee of $11$ members sits a round table so that the President and the Secretary sit together $= 2 \times 9!$ ways