Question

Number of ways in which $3$ boys and $3$ girls can be seated in a row where two particular girls do not want to sit adjacent to a particular boy is_____.

Answer 1

Answer: 288

Let $G_1$ and $G_2$ be the girls who do not want to sit with boy $B_1$.
$\begin{array}{cccccc}\times & \times & \times & \times & \times & \times \\ 1& 2& 3& 4& 5& 6\end{array}$
Case I: $B_1$ occupies the first or sixth place.
If $B_1$ occupies first place, then $G_1$ and $G_2$ cannot sit at second place.
So, $G_1$ and $G_2$ can be seated at four places $(3,4,5,6)$ in ${}^4{P}_2$ ways.
At remaining three places, two boys and one girl can be seated in $3!$ ways.
So, number of ways in this case $={}^4{P}_2\times 3!=72$
Similarly, if $B_1$ occupies sixth place, number of ways will be $72$ .
Case II: $B_1$ occupies second, third, fourth or fifth place.
If $B_1$ occupies second place, $G_1$ and $G_2$ cannot sit at first or third place.
So, $G_1$ and $G_2$ can be seated at fourth, fifth or sixth place in ${}^3{P}_2$ ways.
At remaining three places, two boys and one girl can be seated in $3!$ ways.
So, number of ways in this case $={}^3{P}_2\times 3!=36$
Similarly, if $B_1$ occupies third, fourth or fifth place number of ways is $36$ .
From above two cases, total number of ways
$=72\times 2+36\times 4=288$