We would like to count how many ways 3 boys and 3 girls can sit in a row. How many ways can this be done if:
(b) all the girls sit together? ...
(c) every boy sits next to at least one other girl?
ANSWER: If three boys sit next to each other, no combinations work. If two boys sit next to each other, it fails if and only if the pair of boys sitting next to each other are on an edge (ie. BBGGBG, BGGGBB). If no two boys sit next to each other, all combinations work. There are 4!3! combinations with three boys together (see part (b)). If we place two boys on the edge, we have two choices, left or right to place them. We then choose the position of the third boy from threeremaining positions (he can't be next to the two other boys) for a total of 2*3*3!3! positions (3!3! to account for varying positions of unique boys and girls). Since there are 6! total positionings, there are 6!- 4!3!- 2*3*3!3! = 360 positionings where no two boys sit next to each other.
i get where 6!- 4!3! comes from but don't understand where 2*3*3!3! comes from