Question

In the next World Cup of cricket, there will be 12 teams, divided equally into 2 groups. Teams of each group play a match against each other. From each group 3 top teams will qualify for the next round. In this round, each team will play against the other once. Four top teams from this round, where each team will play against each other three.Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next World Cup will be
A. 54
B. 53
C. 38
D. 55

Answer 1

Hint:In order to solve this question, we will use the formula of combination, that is to choose r items out of n, we apply, $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$. Now in the first match each group has 6 teams and 2 will play at a time. So, for the first round, we will choose 2 out of 6 teams for both the groups. Similarly, we will choose for each round and get our answer.

Complete step-by-step answer:
In this question, we have been asked to find the number of minimum matches that will be played among 12 teams in the next World Cup, for a few given conditions. Now, we have been given that for the first round, 12 teams are divided into 2 equal groups, that is, $\dfrac{12}{2}=6$ teams in each group. And each team plays against every other team once. So, we have to select 2 teams out of 6 teams for each match in each group. Now, we know that to choose r items out of n items, we use the formula of combination, that is, $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$. So, for both the groups, the total number of matches for the first round will be,
$2{{\times }^{6}}{{C}_{2}}$
And we can further write it as,
$2\times \dfrac{6\times 5}{2}\Rightarrow 2\times 3\times 5\Rightarrow 30$
So, there will be 30 matches in the first round. Now, we have been given that in the second round, 3 top teams of each group will be taken. So, there will be 6 teams and it is given that each team will play against the other team. So, we will choose 2 teams at a time of the 6 teams. So, the number of matches can be written as,
$^{6}{{C}_{2}}\Rightarrow \dfrac{6\times 5}{2}\Rightarrow 3\times 5\Rightarrow 15$
So, there will be 15 matches in the second round. Now in the third round, 4 top teams are to be selected. So, we can say that in third round, the number of matches for each team against the other will be given as,
$^{4}{{C}_{2}}\Rightarrow \dfrac{4\times 3}{2}\Rightarrow 2\times 3\Rightarrow 6$
Now, in the final round, we have been given that 2 teams will play and best out of three matches will be counted. So, if one team wins for the first and second match, then the third match will not be conducted. So, the minimum matches in the final round will be 2.
So, we can say that the minimum number of matches in the next World Cup will be the sum of the number of matches in the first round, second round, third round and the final round.
So, the number of matches = 30 + 15 + 6 + 2 = 53 matches.
Hence, the minimum number of matches in the next World Cup will be 53 matches. Therefore, option B is the correct answer.

Note: While solving this question, the possible mistake we can make is by ignoring the word, ‘minimum’ which will give an incorrect answer, because in the final round, if the first 2 matches are won by the same team, then the third match will not be conducted as that team will be the winners. So, we have to add 2 and not 3 in the final round to get the correct answer.