1. The football league of a certain country is played according to the following rules:
Each team plays exactly one game against each of the other teams.
The winning team of each game is awarded l point and the losing team gets 0 point.
If a - match ends in a draw, both the teams get 1/2 point.
After the league was over, the teams were ranked according to the points that they earned at the end of the tournament. Analysis of the points table revealed the following:
Exactly half of the points earned by each team were earned in games against the ten teams which finished at the bottom of the table.
Each of the bottom ten teams earned half of their total points against the other nine teams in the bottom ten. How many teams participated in the league?
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By: anil on 05 May 2019 02.37 am
Number of teams in the bottom group = 10
=> Number of matches played among the bottom group teams = $$C^{10}_2$$ = $$frac{10 imes 9}{1 imes 2} = 45$$ Number of points bottom group teams get playing amongst themselves = 45 Let the total number of teams in top group = $$n$$ It is given that bottom teams get 45 points from their matches against top group teams, => 45 out of $$10 n$$ points Number of points that top group teams gets from matches played amongst themselves = $$C^n_2$$ Number of points that top group gets against the bottom group = $$10n - 45$$ => $$C^n_2 = 10n - 45$$ => $$n (n - 1) = 20n - 90$$ => $$n^2 - 21n + 90 = 0$$ => $$(n - 6) (n - 15) = 0$$ If, $$n = 6$$, top group would get = $$C^n_2 + 10n - 45$$ = $$C^6_2 + 60 - 45 = 30$$ Average points per game = $$frac{30}{6} = 5$$ Bottom teams will get on an average = $$frac{45 + 45}{10} = 9$$
This is not possible. => $$n = 15$$ $$ herefore$$ Total number of teams = $$15 + 10 = 25$$
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=> Number of matches played among the bottom group teams = $$C^{10}_2$$ = $$frac{10 imes 9}{1 imes 2} = 45$$ Number of points bottom group teams get playing amongst themselves = 45 Let the total number of teams in top group = $$n$$ It is given that bottom teams get 45 points from their matches against top group teams, => 45 out of $$10 n$$ points Number of points that top group teams gets from matches played amongst themselves = $$C^n_2$$ Number of points that top group gets against the bottom group = $$10n - 45$$ => $$C^n_2 = 10n - 45$$ => $$n (n - 1) = 20n - 90$$ => $$n^2 - 21n + 90 = 0$$ => $$(n - 6) (n - 15) = 0$$ If, $$n = 6$$, top group would get = $$C^n_2 + 10n - 45$$ = $$C^6_2 + 60 - 45 = 30$$ Average points per game = $$frac{30}{6} = 5$$ Bottom teams will get on an average = $$frac{45 + 45}{10} = 9$$
This is not possible. => $$n = 15$$ $$ herefore$$ Total number of teams = $$15 + 10 = 25$$