Question

In a knockout tournament involving 27 teams: Total number of matches to be played is, and Number of byes in the first round is

• A. 25 matches, 5 byes
• B. 26 matches, 5 byes
• C. 26 matches, 3 byes
• D. 27 matches, 5 byes

Hint:

For any knockout tournament, the number of matches is always \( N - 1 \) regardless of whether \( N \) is odd or even.
To find byes, list your powers of 2 (2, 4, 8, 16, 32, 64...) and choose the first value greater than the team count.
Subtracting the team count from this power of 2 gives you the correct number of byes immediately.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks for the total matches and the count of first-round byes in a single knockout tournament with 27 teams.
In a knockout tournament, any team that loses a match is immediately eliminated.
Byes are given in the first round to ensure that the number of teams remaining in the second round is a power of 2, allowing a balanced tournament bracket.

Step 2: Key Formula or Approach:
To solve tournament structure questions, we apply two standard mathematical formulas:
1. Total number of matches (\( M \)) in a knockout tournament with \( N \) teams:
\[ M = N - 1 \]
2. Total number of byes (\( B \)) given in the first round:
\[ B = 2^x - N \]
Where \( 2^x \) is the next power of 2 greater than or equal to the total number of teams \( N \).

Step 3: Detailed Explanation:
1. We start with the total number of teams, given as \( N = 27 \).
2. We calculate the total number of matches to find the tournament winner:
\[ \text{Matches} = 27 - 1 = 26 \text{ matches} \] 3. In a knockout format, each match eliminates exactly one team.
4. To eliminate 26 teams and leave 1 champion, we must play exactly 26 matches.
5. Next, we determine the number of byes for the first round of play.
6. We find the powers of 2: \( 2, 4, 8, 16, 32, 64 \), and so on.
7. The next power of 2 immediately greater than 27 is 32 (which is \( 2^5 \)).
8. Now, we calculate the number of byes using the formula:
\[ \text{Byes} = 32 - 27 = 5 \text{ byes} \] 9. These 5 teams will not play in the first round and will advance directly to the second round.
10. This configuration results in exactly 26 matches and 5 byes.

Step 4: Final Answer:
The calculations show there are 26 matches and 5 byes, which matches Option (B).