Question 79

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 1 point and the losing team gets 0 point. If a - match ends in a draw, both the teams get \frac{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?

Solution

Number of teams in the bottom group $$= 10$$
Let the total number of teams in top group = $$n$$
Total number of teams $$= 10 + n$$

=> Number of matches played amongst the bottom group teams = $$^{10}C_2$$
= $$\frac{10 \times 9}{1 \times 2} = 45$$

Number of points bottom group teams get playing amongst themselves $$= 45\cdot1=45$$

Let the total number of teams in top group = $$n$$

wkt, "Each of the bottom ten teams earned half of their total points against the other nine teams in the bottom ten"
i.e. they get half the total points by playing amongst themselves and the other half of total points by playing with the top group teams.

Since they got 45points playing amongst themselves, the bottom teams get 45 points from their matches against top group teams, => 45 out of $$10 n$$ points
Total points by matches between top teams and bottom teams $$= 10\cdot{n}\cdot1 =10n$$ points

Number of points that top group teams get from matches playing amongst themselves = $$^nC_2$$

Number of points that top group gets against the bottom group = $$10n - 45$$

wkt, "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.". Therefore the top n teams obtained half points by playing amongst themselves and half the points by playing against bottom 10 teams.

=> $$^nC_2 = 10n - 45$$

=> $$n (n - 1) = 20n - 90$$

=> $$n^2 - 21n + 90 = 0$$

=> $$(n - 6) (n - 15) = 0$$

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.

=> $$n = 15$$

$$\therefore$$ Total number of teams = $$15 + 10 = 25$$