Three boys and three girls are to be seated around a circular table. Among them the boy X does not want  any girl as a neighbour and girl Y does not want any boy as a neighbour. The number of possible arrangement is:

Clearly X should sit between the two remaining boys and Y should sit between the remaining two girls We have to arrange two groups B1XB2 and G1YG2 along a circle. This can be done is (2−1)! ways. But group of boys (girls) can be arranged in  2! ways. Thus, the required number of ways is (2) (2) (1) = 4 ways.