Question

11. The AFL (Australian Football League) has 18 teams. Each team plays one game per week. Assume a team never plays a given other team more than once. Prove that after 8 weeks, there exists at least 3 teams none of which have played each other.

Answer 1

1) There are total 18 teams that are playing in the league

2)Each team plays one game per week according to the question

3) So number of games played in one week is 18/2 = 9 games . As two teams plays one match.

4) To choose 3 teams from 18 days there are ¹⁸C₃ ways which equals 816 ways

*18c3 = 18! / 3!(18-3)!

18! = 18 x 17 x 16 x 15..................x1

3!= 3 x 2 x 1

15!= 15 x 14 x 13 x 12........x1

5) For each pair of 3 teams having one game there different combinations is 816 different games

6)But as per questions there are games in each week so total number of games to be played in 8 week is (9x8) = 72 games so its possible that each 3 pair of team play atleast one game

7)Hence atleast 3 teams are there none of which has played each other