In a league cum knockout tournament where 12 groups of 4 teams and 4 groups of 5 teams each are participating. How many matches will be played in league rounds ?

Question

Accepted Answer

To determine the number of matches played in the league rounds of the tournament, we need to calculate the matches for both the groups of 4 teams and the groups of 5 teams.

1. **Groups of 4 Teams:**
   - There are 12 groups, each with 4 teams.
   - In a group of 4 teams, each team plays against every other team once. The number of matches in one group is calculated using the combination formula for selecting 2 teams out of 4 to play a match, which is $ \binom{4}{2} = 6 $.
   - Therefore, for 12 groups, the total number of matches is $ 12 \times 6 = 72 $.

2. **Groups of 5 Teams:**
   - There are 4 groups, each with 5 teams.
   - In a group of 5 teams, each team plays against every other team once. The number of matches in one group is calculated using the combination formula for selecting 2 teams out of 5 to play a match, which is $ \binom{5}{2} = 10 $.
   - Therefore, for 4 groups, the total number of matches is $ 4 \times 10 = 40 $.

3. **Total Matches in League Rounds:**
   - Adding the matches from both types of groups gives us the total number of matches in the league rounds: $ 72 + 40 = 112 $.

Thus, the correct answer is Option 2: 112 matches.

**Explanation for Incorrect Options:**

- **Option 1 (68):** This number is too low and does not account for all the matches in both the 4-team and 5-team groups. It might be a miscalculation of the matches in fewer groups or incorrect group sizes.
  
- **Option 3 (292):** This number is excessively high and likely results from overestimating the number of matches per group or miscalculating the number of groups.

- **Option 4 (144):** This number is higher than the correct total and could result from an incorrect assumption about the number of matches per group or an error in multiplying the number of matches by the number of groups.

Therefore, based on the calculations and logical reasoning, Option 2 is the correct answer, with a total of 112 matches played in the league rounds.