An Australian Open where each player played every other player?

Grand Slam tennis tournaments have a field of 128 players. Why 128? Because it’s the sweet spot in between 64 and 256 ...

If we design a tournament in which every player plays a match against every other player, then every single player would have to play at least 127 matches throughout the tournament (exhausting!) and there would be at least 8,128 matches in total. The tournament would have to run over most of the year!

But how do we come up with 8,128?

A branch of mathematics called combinatorics is used by mathematicians to quickly count things without actually having to count them one by one. For instance, how many tennis matches would need to be played before every player in the field of 128 has played every other player? This is the same as asking how many possible pairs of players we can make by choosing 2 players out of the 128.

Let’s imagine the two players standing next to each other… 

_A_ , _B_

Now we have to select the players. We start by choosing the first player in the pair. How many choices do we have? There are 128 players in the tournament, so we have 128 options for the first player … 

128 , ___

Next, we choose the second player in the pair. How many choices do we have now? We’ve already picked out 1 player, so that leaves 127 players left to choose from … 

128 , 127

Now for each of the 128 choices of the first player, we can make 127 selections of the second player. So, the total number of possible pairs is

128 × 127 = 16,256

But wait, we counted too many pairs! Here’s the problem: choosing Serena Williams first and Sam Stosur second gives us the same pair of players as choosing Sam Stosur first and Serena Williams second. So, we’ve actually counted each pair twice. To reach the correct number of pairs, we have to halve our count …

128 × 127 / 2 = 8,128

which is 8,128. With this method, we can find the answer to our question without having to count each combination individually.

So, instead of each player playing every other player, a knock-out system is used, where players are eliminated from the tournament as soon as they lose a match. This means that the total number of matches we need is 127, which is much more manageable!