CAT 2025 Lesson : Teams & Events - Introduction

1. Introduction

Questions involving teams and events are quite structured and we will get quite a bit of data. These would usually revolve around tournaments and leagues. For instance, we can have a round-robin tournament (where each team plays one match against all the other teams) across multiple rounds, and we could be given each team's score after each round. We could then be provided with some additional information of wins and losses, as well as other clues; to help us identify which teams played in each round, the ranks of the teams, etc.

These questions occur in CAT as well as other entrance tests regularly, but not each year. You can expect 1-2 cases in each year (across all entrance tests).

You may feel that such questions are difficult to solve, but you will find that tackling the clues methodically will help you to solve the case. Let us solve some questions, so that you can learn how to go about these questions.

Knockout Tournaments

There are 32 players participating in a singles knockout tournament. A match is played between 2 players. The winner of a match moves to the next round while the loser exits the tournament.

1) How many matches are played in the tournament?
2) How many rounds are there in the tournament?
3) How many matches does the champion win?
4) If a player won exactly 2 games in the tournament, then which was the match that he lost?

Solution

1) Each match consists of two players and all are knockout matches. A total of 32 players were participating in the tournament. The number of matches in round 1 =

\dfrac{32}{2}

= 16 matches, therefore 16 winners should be in round 1 (No. of matches = No. of winners). These 16 winners of round 1 will be competing against each other in the next round.

The number of matches in round 2 = 8 matches.
The number of matches in the quarter-finals = 4 matches.
The number of matches in semi-finals = 2 matches
The finals round has only one match and the winner of the finals will be the winner of the tournament.

The total number of matches in the whole tournament = 16 + 8 + 4 + 2 + 1 = 31

Alternatively

Concept: Number of Matches = Number of Players – 1

We can also find the total number of matches using the above concept.
Number of matches = 32 – 1 = 31 matches

Answer: 31

2) Each match consists of two players and all are knockout matches. A total of 32 players were participating in the tournament. The number of matches in round 1 =

\dfrac{32}{2}

This tournament has a total of 5 rounds.

Answer: 5

3) Each match consists of two players and all are knockout matches. A total of 32 players were participating in the tournament. The number of matches in round 1 =

\dfrac{32}{2}

This tournament has a total of 5 rounds, therefore a player has to win 5 matches to become the champion of the tournament.

Answer: 5 matches

4) Using the table that we have formed,

If the player had two wins in the whole tournament then he faces a loss in the quarter-finals (the

\bm{3^{rd}}

match).

Answer: Quarter-finals

Answer:
1) 31
2) 5
3) 5
4) Quarter-finals

League Tournaments

In the league stage of a tournament, there are 4 teams - A, B, C and D, in a group. In this stage, every team plays a match against every other team (in their group) exactly once.

1) What is the number of matches played by the 4 teams in the league stage?
2) If C wins all its matches, A loses all its matches and there is at least one drawn match in the league stage, what are the results of each of the matches?

Solution

1) The 4 teams A, B, C, and D played 3 matches each. Two teams played in a match, therefore the total number of matches without repetition is

\dfrac{12}{2}

= 6 matches.
We can also use the combination to find the total number of matches.

Number of matches =

^{4}\mathrm{C}_{2}

\dfrac{4 \times 3}{2}

= 6 matches.

Answer: 6 matches

2) The 4 teams A, B, C, and D played 3 matches each. Two teams played in a match, therefore the total number of matches without repetition is =

^{4}\mathrm{C}_{2}

\dfrac{4 \times 3}{2}

= 6 matches.
'>' represents 'defeated' and '=' represents 'drew with'.

It is given that C won all its matches, therefore C won against A, B, and D.
A lost its matches against B, C, and D.
The match between B and D must be a drawn match, as there was at least one draw.

We can infer from the above table that the total number of wins is always equal to the total number of losses and the number of draws in any tournament will be in 2

k

format, i.e., it will be an even number.

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