Question

$7$ men and $5$ women are sitting in a circle and no two women sit together. In how many ways they can sit ?

Answer 1

First fix the men in circular arrangement, which can be done in $(n-1)!=6!$ ways.

Now, there are $7$ places between the men (i.e. $1$ for each pair of men) and $5$ womens to be seated between the men. That can be done in ${}^7{P}_5$ ways.

$\therefore$ Total no. of ways to sit $=6!\times {}^7{P}_5=720\times \frac{7!}{2!}=360\times 5040=1814400$

Hence, they can sit in $1814400$ ways.