Question

Find the number of ways of arranging four boys and four girls in a circle, so that boys and girls are alternate.

Answer 1

Four boys, say and can be arranged in a circle in $(4-1)!=6$ ways. Four each choice of these ways, there are $4!=24$ choices for girls to arrange them in the circle in the required way (here the number of ways of arranging the girls will not be $(4-1)!$, as positions are named, as $b_1,b_2$ between $b_2$ and $b_3$ etc.). Hence required number of ways is $6\times 24=144$