Solution
Understanding the question
Let us first identify the different sets mentioned in this question,
• The universal set is the players of the team.
• Among them,
o Set A is the people who are batsmen,
o Set B is the people who are bowlers, and
o Set C is the people who are wicket-keepers
o In this question, the number of players who can do neither of the above = 0
Now that we have identified the sets mentioned in the question, let’s understand the information given about them
• We are told that the total number of players in the team = 20
• Among them, there are certain number of wicket-keepers and all of them are either batsmen or bowler or both.
o This implies that there is no player, in the team, who is only a wicket-keeper
o The number of wicket-keepers who can only bat = 3, and
o The number of wicket-keepers who can only bowl = 1
• We are also told that there are atleast 5 players, who can only bat, and atleast 5 players , who can only bowl
• Five players in the team are all-rounders, who can bat and bowl, but are not wicket-keepers
• And we need to find out the maximum number of players, who can do all three things.
Draw the Venn Diagram
Now that we have understood all the information that is given to us, let’s represent this information in a two-set venn-diagram.
Let’s find relationships between different entities in the venn diagram using the information given to us
• In this question, n(U) = n(A or B or C)
• Thus,
o n(U) = n(A or B or C) = a + b + c + d + e + f + g = 20
o We know, c = 0, since there is no player who is only a wicket-keeper
So, a + b + d + e + f + g = 20 ……… (1)
• And, we are given that,
o n(only A and C) = 3, which implies that f = 3
o n(only B and C) = 3, which implies that e = 1
o n( only A and B) = 5, which implies that d = 5
• Substituting these values in equation (1), we get, a + b + g = 20 – 3 – 1 - 5 = 11 ……. (2)
• We are also given that,
o n(only A) ≥ 5 => a ≥ 5
o n(only B) ≥ 5 => b ≥ 5
• In equation (2), a + b + g = 11, ‘g’ represents the number of the players, who can do all three things
o For ‘g’ to be maximum, (a + b) must be minimum
o We know that the minimum value of a and b are 5 each. So, the minimum value of a + b = 5 + 5 = 10
o Therefore, the maximum value of g = 11 – 10 = 1
Hence, the maximum number of players, in the team, who can do all three things = 1
Hence, the correct answer is option B.
Therefore you can have those that can only bat that are wicket or not wicket, and those that can only bowl that are wicket or not wicket.