Dear StackExchange dwellers,
Do you have any idea why this isn't 84? The answer gives: The only way to arrange the five people on the bench, without having two boys seating next to each other is: B-G-B-G-B (when, B-Boy, G-Girl). The number of ways to arrange the students as requested is: (All options) - (This specific option) = 5! - (2! x 3!) = 120 - (2 x 6) = 120-12 = 108. Notice that in that "specific option" the internal order of the Boys and of the Girls may differ, so there are (2! x 3!) options to subtract. In the "specific option": 2! represents the girls' possible arrangements and 3! represents the boys' possible arrangements.
But what about? Ways 3 boys sit next to each other (3!*3!) + Ways 2 boys sit next to each other (4!*2) = 84
Thanks for your time!