Explanation:

There are many types of tournaments held. Some are leagues held in round-robin format, leagues in double round-robin format, knock-outs, round-robin with knock-outs.

The example you stated is a double round-robin league format where each team plays the rest of the team TWICE in the whole course of the league or a ‘season’. So let me EXPLAIN the mathematics behind it:

Say there are 5 teams in a league, let’s name them A,B,C,D and E respectively. So Team A has to play Teams B,C,D and E twice which makes it 8 matches for Team A.

Similarly, Team B has to play Teams A,C,D and E which makes it 8 matches for team B too, but remember we already calculated the 2 matches of A VS B so we need to subtract it here which makes it 6 matches for Team B.

Team C, hence has a count of 4 matches as we already counted 2 matches of A vs C and B vs C.

Extending the logic to Teams D and E, they will have a count of 2 matches and 0 matches respectively.

So adding all of them together, we GET :

8 + 6 + 4 + 2 + 0

Applying the formula for Arithmetic PROGRESSIONS,

S = (n/2)(a + (n-1)*d)

where ,

S = Sum of terms,

n = Number of terms,

a = First term,

d = Common difference

Therefore,

S = (5/2)(0 + (4)*2)

S = (5/2)(4*2)

S = 5 * 4 = 20

So essentially it is S = n * (n-1)

n being the number of teams and S is the number of matches.

Sorry for boring you with the long post but my inner engineer was invoked! :D