Example Case 2

Directions for questions 3 to 7: Answer the following questions based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.
[CAT 2000]

3) What is the total number of matches played in the tournament?

(1) 28
(2) 55
(3) 63
(4) 35

4) The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is

(1) 5
(2) 6
(3) 7
(4) 4

5) What is the highest number of wins for a team in the first stage in spite of which it could be eliminated at the end of first stage?

(1) 2
(2) 3
(3) 4
(4) 5

6) What is the number of rounds in the second stage of the tournament?

(1) 1
(2) 2
(3) 3
(4) 4

7) Which of the following statements is true?

(1) The winner will have more wins than any other team in the tournament.
(2) At the end of the first stage, no team eliminated from the tournament will have more wins than any of the teams qualifying for the second stage.
(3) It is possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the first stage.
(4) The number of teams with exactly one win in the second stage of the tournament is 4.

Solution

3) The first stage of the ABC Gold Cup cricket tournament has two groups represented as group 1 and group 2. Each group has 8 teams, and a team plays every other team in its group exactly once. Each team in their respective group played 7 matches with other teams in the group exactly once. So a total of $^{8}\mathrm{C}_{2}$ matches are played in a group.

The total number of matches in group 1 and 2 in the first stage of the tournament = $^{8}\mathrm{C}_{2}$ + $^{8}\mathrm{C}_{2}$ = 28 + 28 = 56 matches.

The top four ranked teams from groups 1 and 2 are the teams going to compete in the elimination matches of stage two. Only 8 teams will be playing in stage two of the tournament. Since all the matches in stage two are elimination matches, this stage comprises only three rounds. The 1^st round (Quarter-final) has four elimination matches, the 2^nd round (Semi-final) has two elimination matches and the final round has only one elimination match which decides the winner of the tournament.

The total elimination matches in the stage two of the tournament = 4 + 2 + 1 = 7

Total number of matches played in the tournament = 56 +7 = 63 matches

Answer: (3) 63

4) If a total of 28 matches are played in a group, then there should be 28 wins in that particular group.

To find the guaranteed minimum number of wins needed for a team to move to the next stage, we have to maximise the number of wins of a particular team that does not advance to the next stage. Let us take the teams A, B, C, D, E, F, G, and H are ranked according to alphabetical order. Each played 7 matches with other teams in the group exactly once.

The top four teams moved to the next round are A, B, C, and D. So we should maximise the number of wins of team E which ranked 5^th in the group. To maximise the wins of E we should minimise the wins of F, G, and H.

Average point of 5 teams = $\dfrac{28}{5}$ = 5.6

Initially, we can allocate 5 wins for each of the 5 remaining teams which results in a total of 25 wins. The remaining three wins must be allocated to the top three teams in any order, i.e., a total of 7 wins for A and 6 wins for B or 6 wins for A, B, and C.

So a team needs a minimum of 6 wins to advance to the next stage of the ABC Gold Cup cricket tournament.

Answer: (2) 6

5) If a total of 28 matches are played in a group, then there should be 28 wins in that particular group.

Let us take eight teams in a group A, B, C, D, E, F, G, and H are ranked according to the alphabetical order. Each played 7 matches with other teams in the group exactly once.

The top four teams moved to the next round are A, B, C, and D. So we should maximise the number of wins of team E which ranked 5^th in the group. To maximise the wins of E we should minimise the wins of F, G, and H.

Average point of 5 teams = $\dfrac{28}{5}$ = 5.6

Initially, we can allocate 5 wins for each of the 5 remaining teams which results in a total of 25 wins. The remaining three wins must be allocated to the top three teams in any order, i.e, a total of 7 wins for A and 6 wins for B or 6 wins for A, B, and C.

E can get a maximum of 5 wins and can still be eliminated in the 1^st stage.

Answer: (4) 5

6) The first stage of the ABC Gold Cup cricket tournament has two groups represented as group 1 and group 2. Each group has 8 teams, and a team plays every other team in its group exactly once. Each team in their respective group played 7 matches with other teams in the group exactly once. So a total of $^{8}\mathrm{C}_{2}$ matches are played in a group.

The total number of matches in group 1 and 2 in the first stage of the tournament = $^{8}\mathrm{C}_{2}$ + $^{8}\mathrm{C}_{2}$ = 28 + 28 = 56 matches.

The top four ranked teams from groups 1 and 2 are the teams going to compete in the elimination matches of stage two. Only 8 teams will be playing in stage two of the tournament. Since all the matches in stage two are elimination matches, this stage comprises only three rounds. The 1^st round (Quarter-final) has four elimination matches, the 2^nd round (Semi-final) has two elimination matches and the final round has only one elimination match which decides the winner of the tournament.

Therefore, the 2^nd round of the tournament has 3 rounds.

Answer: (3) 3

7) Let us take two groups, groups 1 and 2 having 8 teams each to check the condition in the 1^st option.
Group 1 - A, B, C, D, E, F, G, and H
Group 2 – P, Q, R, S, T, U, V, and W

In option 1, it is given that the winner will have more wins than any other teams in the tournament. In the following tables assume A, B, and C has 7, 6, and 5 points, and the remaining teams have 2 points each. Let's assume the same values for the teams in group 2.

A, B, C and D are the four teams that will be competing with the other top four teams (P, Q, R and S) of Group 2. To this condition, let us take two teams – one with minimum points and the other with maximum points – qualified for the 2^nd stage of the tournament. P has a maximum of 7 wins and D has a minimum of 2 wins qualified for the 2^nd stage.

If D wins against P in the final round of stage two, it will have a total of 5 points or wins at the end of the tournament which is less than the number of wins of P, Q, A and B. So the winner doesn't need to have the most number of wins at the end of the tournament.

Option (1) is wrong.

The different assumed wins of each team in the following two tables can be used to check option (2).

From the details in these tables, team T with 5 wins from group 2 is not eligible for the 2^nd stage, but team D which is 3 points less than team T is selected for the 2^nd stage of the tournament. So option (2) is also wrong.

Option (4) is wrong, because a team that wins an elimination match in the 2^nd stage of the tournament will move for the next round and compete with the other winning teams from round one of stage 2. So the number of teams with exactly one win in the second stage of the tournament is 2.

If team D wins the tournament, with respect to the information given in the above two tables, the points of D and T will be the same at the end. So option (3) is correct.

Answer: (3) It is possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the first stage.

Answer:
3) (3) 63;
4) (2) 6;
5) (4) 5
6) (3) 3;
7) (3) It is the possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the first stage.