Question

Three boys and three girls are to be seated around a table in a circle. Among the boys, $X$ does not want any girl neighbour and the girl $Y$ dries not want any boy neighbour. How many such arrangements are possible?

Answer 1

$X$ does not want any girl neighbour and the girl $Y$ dries not want any boy neighbour.

Therefore,

$X$ and $Y$ have their seats fixed. The remaining boys can be seated in $2!$ Ways and similarly girls in $2!$ Ways

Total $2\times 2=4$