Problem-Ways to sit on chairs

Hi!!!I have a question.
There are n girls n boys and 2n chairs in a row.With how many ways can they sit on the chairs so that next to each girl sits a boy and next to each boy sits a girl??

In other words, you must alternate boys and girls.

1) Choose to start with either a girl or a boy in the first seat. There are, of course, 2 ways to do that.

2) Choose one of that gender to seat first. There are n ways to do that.

3) Choose one of the opposite gender to seat next. There are n ways to do that.

4) Choose one of the first gender to seat third. There are n-1 ways to do that.

5) Choose one of the opposite gender to seat fourth. There are n-1 ways to do that.

Continue like that down to "2", "2", "1", "1". The total number of ways to do this is 2(n)(n)(n-1)(n-1)(n-2)(n-2)...(2)(2)(1)(1)= 2(n!)(n!).

Ok...I got it!!!Thank you very much!!!

HallsofIvy said:

In other words, you must alternate boys and girls.

Click to expand...

To those permutations, you could add

G B B G G B B G G ...

Actually, after you have seated each pair, the next pair is independent of previous pairs, and could be either GB or BG

(GB) (GB) (GB) . . .
where (GB) means either G B or B G

Thank's, Dr Phil. I read this "too fast" and thought "boy, girl, boy, girl" but that would be each boy has a girl on both sides, etc.. "By each girl sits a boy and by each boy sits a girl" only requires opposite sex on ONE side and could be "boy, girl, girl, boy, ....