Q.15 professors have been invited for a round table conference by Vice chancellor of a university. What is the probability that two particular professors occupy the seats on either side of the Vice Chancellor during the conference.

Including the Vice Chancellor, 16 people sit around a round table, giving $(16-1)!=15!$ equally likely circular arrangements. For the favourable event, the 2 particular professors must occupy the 2 seats on either side of the VC: they can be arranged in those 2 seats in $2!$ ways, and the remaining 13 professors fill the other 13 seats in $13!$ ways, giving $2!\tim …

Circular permutation counting: total arrangements are $(n-1)!$ for $n$ people around a table; favourable arrangements fix …

Using $16!$ (linear arrangement count) instead of $15!$ (circular arrangement count) for the total sample space, or forgetting the VC's …