In a tournament with $7$ teams, each team plays one match with every other team. For each match, the team earns two points if it wins, one point if it ties, and no points if it loses. At the end of all matches, the teams are ordered in the descending order of their total points (the order among the teams with the same total are determined by a whimsical tournament referee). The first three teams in this ordering are then chosen to play in the next round. What is the minimum total number of points a team must earn in order to be guaranteed a place in the next round?
We have $7$ team and each team will be play $6$ matches. Possible scenario for the maximum marks of third team:
First team won $4$ matches with last $4$ teams and two ties with second and third team then total possible marks of first $=4\times2+1+1=10$
Second team won $4$ matches with last $4$ teams and two ties with first and third team then total possible marks of second $=4\times2+1+1=10$
Third team won $4$ matches with last $4$ teams and two ties with first and second team then total possible marks of first $=4\times2+1+1=10$
Now any of lost $4$ teams can be upto $=3\times2=6$ marks, since last four teams lost three matches each with first, second and third team.
So, maximum marks can be $10$ for third team.
Can you explain in formal way? please.