Question

Eight teams take part in a tournament where each team plays against every other team exactly once. In a particular year, one team got suspended after playing 3 matches, due to a disciplinary issue. The organizers decide to proceed, nonetheless, with the remaining matches. The total number of matches that were played in the tournament that year is ________

Hint:

For tournament problems, there are often multiple ways to count the matches. Calculating the total and subtracting the exceptions is often as effective as adding up the different components. Choose the method that seems more straightforward to you.

Answer 1

Correct Answer: 24

Step 1: Understanding the Concept:
This is a combinatorics problem related to round-robin tournaments. We can solve this by calculating the total matches that would have been played and subtracting the matches that were cancelled due to the suspension.
Step 2: Key Formula or Approach:
The number of matches in a round-robin tournament with 'n' teams is given by the combination formula \( \binom{n}{2} = \frac{n(n-1)}{2} \).
Step 3: Detailed Explanation:
Method 1: Subtracting Cancelled Matches
First, calculate the total number of matches that would have been played if no team was suspended. With 8 teams, this is: \[ \text{Total potential matches} = \binom{8}{2} = \frac{8 \times (8-1)}{2} = \frac{8 \times 7}{2} = 28 \text{ matches} \] The suspended team was scheduled to play against 7 other teams, meaning it had 7 matches scheduled.
The team played 3 matches before being suspended.
The number of matches involving the suspended team that were cancelled is: \[ \text{Cancelled matches} = (\text{Scheduled matches for the team}) - (\text{Played matches}) = 7 - 3 = 4 \text{ matches} \] The total number of matches actually played is the total potential matches minus the cancelled matches. \[ \text{Played matches} = 28 - 4 = 24 \] Method 2: Summing Played Matches
Alternatively, we can sum the matches played by the suspended team and the matches played among the other teams. The suspended team played 3 matches.
The remaining 7 teams continued the tournament among themselves. The number of matches among these 7 teams is: \[ \binom{7}{2} = \frac{7 \times (7-1)}{2} = \frac{7 \times 6}{2} = 21 \text{ matches} \] The total number of matches played is the sum of the matches played by the suspended team and the matches played by the remaining teams. \[ \text{Total played matches} = 3 + 21 = 24 \] Step 4: Final Answer:
The total number of matches that were played in the tournament is 24.