Tennis - Winning a 5 set match

I read this earlier, at The Atlantic in a story about the possible demise of 5 set matches at grand slams.

A best-of-three format (say, Player A vs. Player B) can produce only four distinct three-set outcomes: ABA, ABB, BAA and BAB. A five-set match, on the other hand, has 12 possible set outcomes, creating a host of ways momentum can twist and turn over the course of several hours.

I hadn’t thought about this before.

The final sentence seemed a little odd, until I realised they were talking about the number of outcomes when then match goes to the full five sets.  They are deliberately ignoring the matches that only last 3 or 4 sets.

Anyway - this is more interesting than the common scenario when we want to know how many arrangements there are if you have two things to choose from.  With five sets in a match (and using Player A and Player B as in the Atlantic article) you would get 2^5 = 32 outcomes.  Some of these, of course, are not real, because if Player A wins all of the first three sets (a straight sets victory) the game ends without the final two sets being played, regardless of whether Player A would have gone on to wins them or whether Player would have done.

Asking secondary pupils to work out how many different ways there are to win a five-set match (when it doesn’t have to go the full distance) seems like a nice idea.  I can think of three ways they might set (!) this out.  Are there more?

One is to use a tree diagram and to cross out the ones that will never crop up because one player has already won.  The bottom row shows the 12 ways mentioned in the original article. 

There are therefore (from this diagram) 20 different possible ways for the match to finish.

 

Another is to make a list of these outcomes and not to count the ones that are the same as other matches that have already finished.  The numbers in the next diagram show the counting, while the shading shows who has won:

The symmetry here is nice and shows that I could have got away with just doing the top half and then doubling the number of ways.  This is just a repeat of the tree diagram, though, so the next way concentrates more on the combinations that work.

Here are the ways that the game can progress if player A wins the first two sets:

If Player A wins the first set and B wins the second then this can happen:

…which gives 10 ways so far.  Then we can exchange A and B to give us the other 10 ways.