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A total of 15 teams participated in a tournament. Each team plays with every other team exactly once. A team gets 3 points for a win, 2 points for a draw and 1 point for a loss. The team which scored the least got 21 points. The scores of all the teams were distinct and at least one match played by winning team was drawn. Which of the following is always true for the winning team?


It had at least two draws


It had maximum of four loss


It had a maximum of 9 wins


All of the above

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Option(B) is correct

Number of points scored by any team when it plays with another =4. 

Total number of match played = 105

Total number of points of all the matches played = 420.

As the losing team scored 21 points and the number of points scored by the teams are distinct integers, only if their scores are consecutive.

The total number of points of all matches played would be 420. 

Hence winning team has scored 35 points.

Let the number of matches won, drawn and lost by the winning team be $W,D$ and $L$ respectively.

⇒ $3W+2D+L=35$ and: 


Subtracting the second eq from the first eq
We get: 


Also, $W+D≤14$

(W,D) can be (10,1) or (9,3) or (8,5) or (7,7)

The respective value of $L$ is 3 or 2 or 1 or 0.

(3) Comment(s)


how did this come Total number of points of all the matches played = 420.


Number of matches played by winning team is 14 as each team plays exactly once with every other team.


How is W+D+L = 14. Please someone explain.