Question

There are $4$ boys and $4$ girls. In how many ways they can sit in a row
(i) there is no restriction.
(ii) not all girls sit together.
(iii) no two girls sit together.

Answer 1

(i)-------- ${}^8{P}_4{.}^4{P}_4=\frac{8!}{4!}.\frac{4!}{0!}=8!$

(ii) cases where all gets sit together

$B_1,B_2,B_3,B_4,G$ 5! ways

$G:G_1,G_2{G}_3{G}_4$ 4! ways

$5!\ast 4!$

not all girls together $=(8!-5!4!)$ ways

(iii) $-B_1-B_2-B_3-B_4-$

Boys 4! ways

4 Girls in 5 places : ${}^5{P}_4$ ways

$5!\ast 4!$ ways