Question

In how many ways $6$ persons can sit at a round table, if two of them prefer to sit together$?$

Answer 1

Let $P_1,P_2,P_3,P_4,P_5,P_6$ Pe be the persons, where $P_1,P_2$ want to sit together.
Regard these person as $5$ objects. They can be arranged in a circle in $(5-1)!=24$ ways. Now $P_1,P_2$ can be arranged in $2!$ ways. Thus the total number of ways$=24\times 2=48.$