Directions (Qs. 1-5): Answer the questions based on the following information.
Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament was conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprised several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.
The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage, teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.
Question 1 : The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is :
Question 2 : What is the total number of matches played in the tournament?
Question 3 : Which of the following statements is true?
(a) The number of teams with exactly one win in the second stage of the tournament is 4. (b) The winner will have more wins than any other team in the tournament.
(c) It is possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the first stage.
(d) At the end of the first stage, no team eliminated from the tournament will have more wins than any of the teams qualifying for the second stage.
Question 4 : What is the highest number of wins for a team in the first stage in spite of which it would be eliminated at the end of first stage?
Question 5 : What is the number of rounds in the second stage of the tournament?
Answers and Explanations
Answer 1 : (B)
Since there are 8 teams in a group and each team has to play with every other team in the group exactly once, therefore the number of matches played are [8C2] i.e. 28 matches. And every match has a winner therefore there are 28 wins and similarly 28 losses. To advance to the next stage only 4 teams are selected from a group.
Suppose a team wins 5 matches and similarly there may be a case where 5 teams can win 5 matches each making it 25 wins and since only 4 teams could be selected, a team winning 5 matches cannot be sure of its advancement. However, if a team wins 6 matches it is not possible for more than 4 teams to win 6 matches (as max wins can be 28). Hence it would be assured of its advancement.
Alternative Answer: The following table shows the maximum number of matches won under extreme conditions. Sign (V) denotes win sign(×) denotes defeat and sign (√) denotes a combination of team with itself (no match can be played).
From the table, we find that each of A, B, C, G and H can win 5 matches hence it cannot be decided that which team will qualify for the final. Therefore, the minimum number of wins that can assume a place in the second stage is 6.
Answer 2 : (C)
There will be eight teams in each group. Each team in a group will play with every other team. Hence, total number of matches will be in one group. Hence, in both the groups, there will be 56 matches. This is for the first stage game.
Again, there are 8 teams in knockout rounds from which one winner emerges or 7 losers are identified.
Hence, 7 more matches will be played. So, total number of matches played = (56 + 7) = 63.
Answer 3 : (C)
On the basis of the information, option (c) is the only true statement.
Answer 4 : (D)
The highest number of wins for a team is 4.
Answer 5 : (C)
Since there are 8 teams, there would be 7 matches in 3 rounds.
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