A committee of $15$ members sit around a table . In how many ways can they be seated if the president and vice president sit together ?
Hint : Consider the president and vice president as one member
Consider president and vice president as one member .
Their arrangement in their own group $ = 2!$ (Since they also exchange their seat between each other)
As we know the total members $ = 14$
We know the arrangement of $n$ number of people around a table $=(n-1)!$
Therefore arrangement of 14 members around a table $ = 13!$
Total no. of ways in which committee of 15 members can sit around a table including President & Vice-President sit together $ = 13!\, \times 2!$
Answer is $13!\, \times 2!$.
Note :- In this type of question of arranging people around a table, we should keep in mind that if we want to arrange n number of items around a table then the number of arrangements are (n-1)!.Here we have a condition that the President & Vice-President will sit together therefore modifications occurred . We have multiplied 2! because we have considered them as one member but they can exchange their place between themselves.