Find the number of ways in which 3 boys and 3 girls can be seated on a line where two particular girls do not want to sit adjacent to a particular boy.

Explanation

Let `B_(1)` be the boy and `G_(1),G_(2)` be the girls who do not want to sit with `B_(1)`.
`{:(xx,xx,xx,xx,xx,xx),(1,2,3,4,5,6):}`
Case I : `B_(1)` occupies the first or sixth place
If `B_(1)` occupies first place then `G_(1) " or" G_(2)` cannot sit on second place
`therefore G_(1) " and" G_(2)` can be arranged in four places (3,4,5,6) in `.^(4)P_(2)` ways.
In remaining three places two boys and one girl can be arranged in 3! ways.
So, number of ways in this case `= .^(4)P_(2)xx3!=72`
Similarly if `B_(1)` occupies sixth place, `G_(1) " and" G_(2)` cannot sit on first or third place.
So, `G_(1) " and" G_(2)` can be arranged in fourth, fifth or sixth place in `.^(3)P_(2)` ways.
In remaining three places two boys and one girl can be arranged in 3! ways.
So, number of ways in this case are `.^(3)P_(2)xx3!=36`.
Similarly, if `B_(1)` occupies third, fourth of fifth place number of ways are 36.
From above two cases total number of ways
`=72xx2+36xx4=288`