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Answer the next 10 questions based on the information given below:
16 teams taking part in a cricket tournament are divided into two pools with eight teams in each pool. The teams in a pool play one match each against all other teams in the same pool. Two points are awarded for a win and zero points for a loss. There are no draws or ties in the tournament.
At the end of all the matches in the pool stage, the top two teams in each pool, in terms of the points scored, advance to the semifinals.
In the semifinal, the top team from the first pool plays the second team from the second pool and vice versa. The winners of the semifinals play the final, to determine the winner of the tournament.
If two or more teams end up with the same number of points at the end of the pool stage, the team with better net run rate is placed higher.
How many matches are played in each pool?
Answer: B
As there are eight teams in a pool, the number of matches = 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28.
A → B, C, D, E, F, G and H, i.e., 7 matches
B → C, D, E, F, G and H, i.e., 6 matches
C → D, E, F, G and H, i.e., 5 matches
D → E, F, G and H, i.e., 4 matches
E → F, G and H, i.e., 3 matches
F → G and H, i.e., 2 matches
G → H, i.e., 1 match
Alternately,
Number of matches in each pool = 8C2 = (8 × 7)/2 = 28.
Hence, option (b).
How many matches are played in the tournament?
Answer: D
The total number of matches = 28 + 28 + 2 + 1 = 59
⇒ Pool matches = 28 + 28,
⇒ Semifinals = 2
⇒ Final = 1
Hence, option (d).
What is the minimum number of points with which a team can reach the semifinals?
Answer: A
Let A and B advance to next round with B qualiying with least possible score.
To minimise the points scored by B we will have to maximise the points of the teams above it i.e., of A. A plays a total of 7 matches hence maximum points A scores will be 14.
Total points to be score in the pool stage = 28 × 2 = 56.
Remaining points = 56 – 14 = 42.
Now, these points are to be distributed amongst B to H such that B who is the best of these 7 teams gets least number of points.
The best team cannot receive less than the average, hence points when shared equally by the seven teams, each team would get 42/7 = 6 points.
Since all 7 teams have same points, B can advance based on net runrate.
∴ A team with six points can advance to the semifinals.
Hence, option (a).
What could be the maximum points scored by a team that failed to reach the semifinals?
Answer: A
If three teams end up with the maximum points, then one of them would be eliminated.
Total points to be score in the pool stage = 2 × 28 = 56.
Now, bottom 5 teams will play = 5C2 = 10 matches between them i.e., they will get minimum 10 × 2 = 20 points between them.
Now, maximum total points that the top 3 teams get is 56 – 20 = 36.
Let’s distribute these 36 points equally among the top 3 teams. Based on net run-rate 2 of these 3 teams will advance and 3rd team will be eliminated.
So, maximum score a team can get when three teams finish with highest points is 12.
∴ A team with 12 points can be eliminated.
Hence, option (a).
What could be the maximum number of points scored by a team that finished last in a pool at the end of the pool stage?
Answer: C
If all teams score the same number of points, each team will have 56/8 = 7 points, but as there are no draws and ties, some teams would score 8 and some teams would score 6 points.
A team with 6 points can end up as last in the pool.
Hence, option (c).
Answer the next 2 questions based on the information given below.
There are 16 teams and they are divided into 2 pools of 8 teams each. Each team in a group plays against every other team on a round-robin basis. Draws in the competition are resolved based on extensive rules. The top four teams from each group will qualify for the next round i.e. round 2. In case of teams having the same number of wins, the team with a better net run-rate would be ranked ahead.
What is the minimum number of wins required to guarantee qualification to the next round?
Answer: 6
Instead of calculating the minimum number of wins required to guarantee qualification to the next round, let us focus on maximum numbers of wins despite a team can be eliminated.
Total number of matches in a pool = 8C2 = 28 matches.
Now, 4 teams will qualify for next round and the 5th team despite wining a lot of matches would get disqualified.
The bottom 3 teams will play 3 matches.
∴ The top 5 teams will play a total of 28 - 3 = 25 matches i.e., they can win a maximum of 25 matches.
Now to maximise the number of matches team 5 wins, we will distribute these 25 wins equally amongst all 5 teams. Now there are 5 teams with 5 wins each. Since only 4 teams can qualify there will be one team that would get disqualified.
Hence, even after winning 5 matches a team can get disqualified.
∴ To gurantee qualification to next round a team will have to win at least 6 matches.
[Note: A team may be able to qualify even with less than 6 wins, but that would not be a guaranteed qualification.]
Hence, 6.
What is the minimum possible number of wins with which a team can qualify to the next round?
Answer: 0
Since draws are allowed, it is possible that all 28 matches in the pool are drawn and based on net runrate top 4 teams will qualify to next round.
Hence, 0.