Five boys and four girls are to be seated in a row so that two particular girls will never sit adjacent to a particular boy and all girls are separated. If the number of ways in which they can be seated is N, the value of #N/(480)# is?

If I'm reading this right, we have 5 boys and 4 girls and are seated in a row such that the girls are all separated:

#Bcolor(white)(0) Gcolor(white)(0) B color(white)(0)G color(white)(0)Bcolor(white)(0) Gcolor(white)(0) Bcolor(white)(0) Gcolor(white)(0) B#

With no other restrictions, the number of ways we can arrange this grouping is:

#N=5!xx4! = 120xx24=2880#

At this point, #N/480=6#

For the language about 2 particular girls not sitting next to a particular boy, we can put that restriction in. Let's first freeze the two girls (they'll be the first two Gs). We don't want the boy to be the second B:

#Bcolor(white)(0) color(orange)Gcolor(white)(0) color(red)B color(white)(0)color(orange)G color(white)(0)Bcolor(white)(0) Gcolor(white)(0) Bcolor(white)(0) Gcolor(white)(0) B#

There are 3 positions along the row that the two girls can sit.

There are 2 ways to arrange the two girls in those seats.

There are 2 ways that the other two girls can sit in the remaining 2 seats. This gives the number of ways to arrange the girls is:

#3xx2xx2=12#

For the boys, the first boy can only sit in one of 4 seats. The remaining boys can sit in any of the remaining 4 seats, giving:

#4xx4! = 4xx24=96#

And so we get:

#N=12xx96=1152=>N/480=2.4#