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 Concept:
In tournament fixtures and sports management planning, a knockout or elimination tournament works on the rule that a team is eliminated immediately after a single loss. - Total Matches: Every match played results in exactly one losing team being eliminated from the competition. To find a single ultimate winner, every team except the champion must lose exactly once. - Byes: A "bye" is a preferential scheduling slot awarded to certain teams in the opening round, allowing them to advance automatically to the second round without playing. Byes are necessary whenever the total number of competing teams is not a perfect power of 2 ($2, 4, 8, 16, 32, 64, \dots$). This adjustment ensures that the remaining field size scales perfectly to a power of two for all subsequent rounds.
Step 2: Key Formula or Approach:
1. Formula for total matches ($N_m$): $$ N_m = N - 1 $$ Where $N$ is the total number of participating teams. 2. Formula for total number of byes ($N_b$): $$ N_b = 2^x - N $$ Where $2^x$ is the next higher power of 2 immediately following the total team count $N$.
Step 3: Detailed Explanation:
Let's substitute our total team count ($N = 27$) into our mathematical tournament formulas: 1. Calculate the total matches to be played: $$ N_m = 27 - 1 = 26 \text{ matches} $$ A total of 26 matches must be organized to eliminate 26 teams and crown 1 champion. 2. Calculate the number of byes for the first round: Identify the powers of 2 surrounding our team count of 27: $$ 16<\mathbf{27}<32 $$ The power of 2 that immediately follows 27 is $32$ (which is $2^5$). Subtract the total team count from this value: $$ N_b = 32 - 27 = 5 \text{ byes} $$ Therefore, a tournament with 27 teams requires 26 total matches and features 5 byes in the first round. This matches option (B).
Step 4: Final Answer:
The total number of matches is 26, and the number of byes is 5.