sorry could you explain further how to get P(Y) case 1 where 4 boys are tgt

P(Y) case 1 is: First arrange the 5 girls in 5! ways. Then the four of the five boys must be arranged in 5P4 ways as I only want four of them first. As the group of four-boys must not be next to the remaining 1 boy, I have to use the slot-in method.

Imagine, you space the five girls apart.
__ G1 __ G2 __ G3 __ G4 __ G5 __
Including the two slots at both ends, there are six empty slots. First I put in the group of 4-boys, there are 6 choices. Next, the remaining boy has 5 slots to choose from. So these are the 6*5 or 6P2 ways.

oh nice! i understand now, thank you so much :D

No worry. I am gladed too!:)