## Love triangles

My wife is a big fan of the TV medical drama, Grey’s Anatomy.  I don’t care much for the show myself, but by all accounts it’s a quality production and a hit with audiences.  Besides, it makes my wife happy.  And a happy wife equals a happy life.  As far as I can tell, from my limited exposure, Grey’s Anatomy centres its drama around a series of love triangles.  No, I’m not talking about ménages à trois, so stop that sniggering.  I’m talking about that other kind of messy relationship structure.  You know… boy1 loves girl1 but girl1 secretly pines for boy2 who is married to girl2 but is unfaithful with girl3…  And so on, ad nauseam.

Undoubtedly this makes for great TV.  But I was thinking about it from a statistical/probability point of view.  I wondered just how long could the writers of Grey’s Anatomy keep these shenanigans up?  Exactly how many love triangles are possible?

Let’s assume for simplicity that old characters don’t leave and new characters aren’t introduced.  That is, consider a closed system of m males and f females.  Utilising the combination formula there are m fC2 combinations of male-female-female love triangles and f mC2 combinations of female-male-male love triangles.  And for the sake of completeness there are mC3 male-male-male combinations and fC3 female-female-female combinations.  Not that there’s anything wrong with that.

Looking at the photo above, I count five male doctors and four female doctors.  Assuming that the writers of Grey’s Anatomy create one episode per love triangle combination then that line up of cast can generate:

= m fC2 + f mC2 + mC3 + fC3

= 5 4C2 + 4 5C2 + 5C3 + 4C3

= 84 unique love triangles.

Interestingly, this figure is remarkably close to the 93 episodes actually created so far.  Looking at the statistics, Grey’s Anatomy has simply exhausted itself and there will be no Season 6.

My wife will be devastated!