In how many ways can $15$ members of a council sit along a circular table, when the secretary is to sit on one side of a chairman and deputy secretary on the other side? 

Given :  $15$ members of a council sit along a circular table

To find : no of ways in which the secretary is to sit on one side of a chairman and deputy secretary on the other side. 

A permutation is a (possible) rearrangement of objects. Since total members are $15,$but three special members constitute one member. 

By using the formula $n! =1. 2.3 .... (n-1) . n$

$\therefore$ Required number of arrangements are  $12! \times 2$because, chairman remains between the two specified persons and the person can sit in two ways. 

Hence, option $\left(a\right) 12! \times 2$ is correct.