In how many ways can 11 members of a committee sit at a round table

Question:

In how many ways can 11 members of a committee sit at a round table so that the secretary and the joint secretary are always the neighbour of the president?

 

Solution:

First assume the president(P), Joint secretary(JS) and secretary(S) to be 1 members(as shown below)

So there are 9 members, a number of ways to arrange this 9 people is $8 !$ (The formula used (n-1)!)

Now we need to look at the internal arrangement. There are 2 arrangement possible

So total number of arrangement are $(8 !) \times 2=80,640$

 

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