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

Let $G_{1}$ and $G_{2}$ be the girls who do not want to sit with boy $B_{1}$.
$\begin{matrix}\times&\times&\times&\times&\times&\times\\ 1&2&3&4&5&6\end{matrix}$
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
$=72 \times 2+36 \times 4=288$