The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy $B_1$ and a particular girl $G_1$ never is adjacent to each other, is

The number of ways of arranging 5 boys, and 3 girls in a round table is 7!

Number of ways of arranging ${\mathrm{B}}_1, {\mathrm{G}}_1$ are  together.

              $\left({\mathrm{B}}_1, {\mathrm{G}}_1\right)$ along with 2 girls,, 4 boys

               can be arranged in 6!  ways Around a table.

               ${\mathrm{B}}_1, {\mathrm{G}}_1$ can be inter changed in 2 ways,

Number of ways of arranging 5 boys, 3 girls around a table so that ${\mathrm{B}}_1, {\mathrm{G}}_1$ not together 

$\begin{array}{rcl} & =& 7!-2(6!)\\ & =& 5(6!)\end{array}$