Comprehension
Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is 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 comprises of several rounds. A round involves one match for each team. The winner of a match in a round advance 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
What is the total number of matches played in the tournament?
What is the total number of matches played in the tournament?
Show Hint
In round robin format, matches $= \binom{n}{2}$. In knockout format, matches $= n - 1$.
Updated On: Aug 5, 2025
- 28
- 55
- 63
- 35
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The Correct Option is C
Solution and Explanation
Stage 1: Each group has $8$ teams. Matches in one group = $\binom{8}{2} = 28$.
For $2$ groups: $28 \times 2 = 56$ matches.
Stage 2: $8$ teams in knockout $\Rightarrow$ matches = $8 - 1 = 7$.
Total matches = $56 + 7 = 63$.
\[
\boxed{63}
\]
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Question: 2
The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is:
The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is:
Show Hint
In group stages, guarantee qualification by securing more wins than the maximum possible for the 5th ranked team.
Updated On: Aug 5, 2025
- 5
- 6
- 7
- 4
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The Correct Option is B
Solution and Explanation
Each team plays $7$ matches in stage 1. To ensure top 4 place:
Worst case — multiple teams tie. A record of $5$ wins could still cause a tie for 4th place.
$6$ wins ensures no more than $3$ teams can exceed your wins.
\[
\boxed{6}
\]
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Question: 3
The highest number of wins in the first stage with which a team can still be eliminated is:
The highest number of wins in the first stage with which a team can still be eliminated is:
Show Hint
Check for elimination thresholds by constructing tie scenarios with many teams having equal wins.
Updated On: Aug 5, 2025
- 1
- 2
- 3
- 4
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The Correct Option is D
Solution and Explanation
In an 8-team group, it is possible for $5$ or more teams to have $4$ or more wins, so a $4$-win team could be ranked 5th or lower by tie-breakers.
$5$ wins cannot be eliminated.
\[
\boxed{4}
\]
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Question: 4
What is the number of rounds in the second stage of the tournament?
What is the number of rounds in the second stage of the tournament?
Show Hint
In knockout stages, rounds $= \log_2(n)$ when $n$ is a power of $2$.
Updated On: Aug 5, 2025
- 1
- 2
- 3
- 4
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The Correct Option is C
Solution and Explanation
$8$ teams in knockout:
Round 1: 8 $\to$ 4 teams
Round 2: 4 $\to$ 2 teams
Round 3: 2 $\to$ 1 champion \[ \boxed{3} \]
Round 2: 4 $\to$ 2 teams
Round 3: 2 $\to$ 1 champion \[ \boxed{3} \]
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Question: 5
Which of the following statements is true?
Which of the following statements is true?
Show Hint
Compare total possible wins across both stages to check for equality scenarios between eliminated teams and the champion.
Updated On: Aug 5, 2025
- The winner will have more wins than any other team in the tournament.
- At the end of the first stage, no eliminated team will have more wins than any team qualifying for the second stage.
- 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.
- The number of teams with exactly one win in the second stage of the tournament is 4.
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The Correct Option is C
Solution and Explanation
Winner's possible wins = up to $4$ in stage 1 + $3$ in stage 2 = $7$.
A team in stage 1 could also win $7$ matches but be eliminated in stage 2 or by tie-break in stage 1.
So they could match the winner's total wins.
\[
\boxed{\text{Statement 3 is correct.}}
\]
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