Problem 145 - A Complete Resource Book in Mathematics for JEE Main

Question

In the World Cup, the tournament is arranged as per the following rules: In the beginning, 16 teams are taken and divided into 2 groups of 8 teams each. Teams of each group play a match against each other in the same group. From each group, 4 top teams qualify for the next round. In the next round, two teams play each other in each group and the losing team goes out of the tournament. Then, four winning teams play for semifinal round and finally there is one final. The rules of the tournament are such that every match can result only in a win or a loss and not in a tie. The total number of matches played in the tournament is (A) 51 (B) 64 (C) 63 (D) 52

Step-by-Step Solution

Answer

The total number of matches played in the tournament is 63.

Step 1: Calculate Group Stage Matches

In the group stage, each group consists of 8 teams. Each team plays against every other team in its group. The formula to calculate the total matches in a group is \( \binom{n}{2} = \frac{n(n-1)}{2} \), where \( n \) is the number of teams. Here, \( n = 8 \). Therefore, the number of matches in one group is \( \frac{8 \times 7}{2} = 28 \). Since there are 2 groups, the total number of group stage matches is \( 2 \times 28 = 56 \).

Step 2: Calculate Knockout Stage Matches

After the group stages, 4 teams from each group qualify. This gives a total of 8 teams for the knockout stage. The knockout falls into stages where each match eliminates one team: quarter-finals, semi-finals, and then a final. For 8 teams: - Quarter-finals: \( 4 \) matches (8 teams reduced to 4)- Semi-finals: \( 2 \) matches (4 teams reduced to 2)- Final: \( 1 \) match to determine the winner. Adding these gives \( 4 + 2 + 1 = 7 \) matches.

Step 3: Calculate Total Matches in Tournament

Sum the matches from the group stage and the knockout stage. The total matches are \( 56 \) (from the group stage) plus \( 7 \) (from the knockout stage), which equals \( 63 \) matches.

Key Concepts

Group Stage Matches Calculation

Understanding how to calculate matches in the group stage of a tournament is essential in sports mathematics. Each team within a group plays against all other teams. For 8 teams in a group, this means each team will have 7 opponents. To find out how many matches there are in total, we use a combination formula.

The formula to calculate the number of matches when every team plays against every other team is given by:
\[\binom{n}{2} = \frac{n(n-1)}{2}\]where \(n\) is the number of teams. In this setting, \(n = 8\), so:

• The matches for one group is: \( \frac{8 \times 7}{2} = 28 \)
• Since there are 2 groups, the total matches are: \(2 \times 28 = 56\) matches.

This step ensures we cover all possible matchups in the group stage.

Knockout Stage Matches Calculation

In knockout stages, matches are structured so that losing one means elimination from the tournament. After group stages, 8 teams proceed to knockout rounds. Calculating the number of matches here is straightforward, following the elimination nature of this stage.

In a knockout tournament with 8 teams:

• Quarter-finals: 4 matches are held, reducing 8 teams to 4.
• Semi-finals: 2 matches are held, cutting down from 4 teams to 2.
• Final: 1 match determines the ultimate winner.

Adding these together gives us: \(4 + 2 + 1 = 7\) matches. In knockout stages, each match is decisive, making the calculation based solely on eliminating half the teams with each round.

Tournament Structure in Mathematics

Mathematical principles can easily describe the structure of a sports tournament. This involves different stages: group stages and knockout stages, each with unique calculation methods and purposes.

• Group Stage: This phase ensures every team has the opportunity to compete against each opponent within its group. Mathematical combinations calculate all matchups, providing a comprehensive competition experience.
• Knockout Stage: This phase eliminates teams systematically until a winner is found. Matches are counted by reducing the number of competing teams by half after each round.

In a complete tournament, the total matches are the sum of group stage matches and knockout stage matches. For our example:
- Total group stage matches: \(56\)
- Total knockout stage matches: \(7\)
So, the total number of matches in the whole tournament sums to \(63\), encapsulating the entire tournament's structure mathematically.