Combinatorics Problem: Assigning Seats for 4 Guys and 4 Girls in a Single Row

[Tony11235]

Suppose you want to assign seats for a single row of 4 guys and 4 girls in such a way that each guy is sitting next to at least one girl and vice versa. How many ways are there to do this?

This is not a hard problem at all, but I am lacking a good outlined approach to solving problems of this sort. Anybody mind sharing their strategies?

 

[0rthodontist]

Well, you can count pairs of girls and guys, taking them together, reasoning that there are 2 ways to arrange each pair, and then afterwards considering uniqueness of people. And there is one exception that takes a few different forms that you can handle separately. But before I tell you more, what have you done?

 

[Tony11235]

Well...there can only be at most two guys or two girls next to each other and there are 5 ways to arrange to arrange them in that manner, although i haven't taken into account how many ways each person can be in that format, 2*4!? And there's more of course.

 

[0rthodontist]

I'm not sure what the "five ways" you are describing are. Here is how I count them:
--if every pair of 2 positions, from the left, contains both a boy and a girl, then I count the total number of ways to have that
That is, I count arrangements of the form WWXXYYZZ where every repeated letter is a boy-girl pair. (so for example BGGBBGBG is one arrangement of that form, with B = boy G = girl because it can be divided into BG GB BG BG).
--and the exceptions to that pattern are where you have BB or GG in one of those four "pair" spots WW, XX, YY, or ZZ. Because the number of spots is so small, this amounts to only a couple cases which you can figure out by hand.

Taking into account how many ways you can arrange boys and girls in a given format goes like this: how many ways can you place the four boys into that format? After that how many ways can you place the four girls into that format? Which rule should you use for combining those ways, multiplication or addition?