Formula to calculate number of matches in a single knockout tournament is :

Question

Accepted Answer

In a single knockout tournament, each match results in one team being eliminated. The tournament continues until only one team remains as the winner. To determine the number of matches required in such a tournament, we can use the formula $N-1$, where $N$ is the total number of teams participating.

### Explanation for Option 4: $N-1$

1. **Logical Reasoning**: In a knockout tournament, every match eliminates one team. Therefore, to find the winner among $N$ teams, we need to eliminate $N-1$ teams. Since each match eliminates exactly one team, the number of matches required is exactly $N-1$.

2. **Example**: Consider a tournament with 8 teams. To determine the winner:
   - Round 1: 4 matches, 4 teams eliminated, 4 teams remain.
   - Round 2: 2 matches, 2 teams eliminated, 2 teams remain.
   - Final Round: 1 match, 1 team eliminated, 1 team remains as the winner.
   - Total matches = 4 + 2 + 1 = 7, which is $8-1$.

### Explanation for Why Other Options are Incorrect

1. **Option 1: $\frac{N-1}{2}$**
   - This formula suggests that the number of matches is half of $N-1$. This is incorrect because it does not account for the fact that each match eliminates only one team. For example, with 8 teams, this formula would suggest 3.5 matches, which is not possible.

2. **Option 2: $\frac{N(N-1)}{2}$**
   - This formula is used to calculate the number of matches in a round-robin tournament, where each team plays every other team once. It is not applicable to knockout tournaments, where teams are eliminated after each match.

3. **Option 3: $\frac{N}{2}$**
   - This formula might suggest the number of matches in the first round of a knockout tournament if $N$ is even, but it does not account for subsequent rounds. It does not provide the total number of matches needed to determine a single winner.

In conclusion, Option 4 ($N-1$) is the correct formula for calculating the number of matches in a single knockout tournament, as it accurately reflects the process of eliminating teams until only one remains.