We can also solve this question without using any traditional counting methods.

Let A, B, C, D, E and F be the six students

Each student plays each of the other students four times.
So, for example, student A plays:
Student B 4 times
Student C 4 times
Student D 4 times
Student E 4 times
Student F 4 times
In other words, student A plays a total of 20 games.

Applying the same logic we can see that student B plays a total of 20 games, student C plays a total of 20 games, etc

There are 6 students altogether.
So the TOTAL number of games played = (6)(20) = 120

At this point, we must recognize that every game has been counted TWICE.
For example, when student A played a game against student B, student A counted that as a game, and student B also counted that is a game.

To account for this duplication, we must divide 120 by 2 to get 60

Answer: 60

Since two players play against each other, just see how many non-repetitive pairs are formed

So basically combine the 6 players in teams of two

So just use \(^nC_r = ^6C_2 = \frac{6*5}{2*1} = 15\)

Each match was played 4 times so just multiply that answer by 4, 15*4 = 60

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