In how many ways can $5$ boys and $5$ girls be seated at a table so that no two boys and no two girls sit side by side?
$5$ boys can be seated at the table in $\;$ $P_5 = 5! = 120$ $\;$ ways
$5$ girls can be seated at the table in $\;$ $P_5 = 5! = 120$ $\;$ ways
Amongst themselves, the boys and girls can be seated in $\;$ $2$ $\;$ ways
$\therefore \;$ Total number of ways in which $5$ boys and $5$ girls can be seated at a table
$= 2 \times \left(P_5\right)^2 = 2 \times 120^2 = 28800$ ways