In a cricket team of 20 players, there are a certain number of wicket-

Hey smlprkh,
Thanks for pointing it out.
We have edited the question.
Thanks and Regards,

Hey Archit3110,
It simply means that at least 5 players only bat and at least 5 players can only bowl.
Since this the minimum number of player that can only bat or bowl, the number of players can be more than 5 also.

Thanks and Regards,

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.

The questions says: "If there are at least 5 players in the team, who can only bat or who can only bowl"

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.

Based on your Venn diagram you should have

a+f>=5
b+e>=5

which become

a+f=a+3>=5
a>=2

b+e=b+1>=5
b>=4

Hi EgmatQuantExpert

Thanks for the question.

I find the following portion from question stem a bit misleading:

"If there are at least 5 players in the team, who can only bat or who can only bowl."


I am interpreting this line as follows....
Atleast 5 players who can "either only ball or only bat" i.e. "only ball + only bat together" >= 5


now in solution you have considered:
"only ball" >= 5
"only bat" >= 5
for this to be true, the question stem should have been clearer in stating that there are at least 5 players who can only bat and at least 5 players who can only bowl.

Please clarify. Also, if my understanding is wrong somewhere, please guide me.

Bunuel if you could give your 2 cents too..

This seemed fairly straightforward to me. Please refer to the image below.
The areas with two roles overlapping have 5,3 and 1 player(s). Now to obtain the number of players who perform all 3 roles, the limiting factor is the least number of players who can play two roles, which in this case happens to be 1, and that is the answer.
EgmatQuantExpert , do you see any issues with this approach?

If there are atleast 5 players in the team, who can only bat or who can only bowl.-

I am interpreting this line as follows....
Atleast 5 players who can "either only ball or only bat" i.e. "only ball + only bat together" >= 5


now in solution you have considered:
"only ball" >= 5
"only bat" >= 5
for this to be true, the question stem should have been clearer in stating that there are at least 5 players who can only bat and at least 5 players who can only bowl.

Please clarify. Also, if my understanding is wrong somewhere, please guide me.

At first, it was a little difficult to work through the verbiage of the question. After understanding the 3 Sets, we have the following Sets:

Set W = Wicket Players

Set X = Batters

Set Y = Bowlers


However, every person who is part of Set W is part of

W + X only

W + Y only

or

All 3: W + X + Y

in other words, there are 0 people part of ---- ONLY W


(a) Find the No. of People who do EXACTLY 2 Things (are part of EXACTLY 2 Sets)

"3 of Wicket can only Bat"

X + W ONLY = 3


"1 of Wicket can only Bowl"

Y + W ONLY = 1


"5 people can Bat + Bowl, but are NOT Wicket"

X + Y ONLY = 5



No. of Unique Elements part of EXACTLY 2 SETS = (X + W only) + (Y + W only) + (X + Y only) = 3 + 1 + 5 =

9 people part of EXACTLY 2 Sets



"There are AT LEAST 5 people on the team who can only Bat or who can only Bowl."

This part of the Question ----- I wish it was written a bit more clearly.

Is there a MINIMUM 5 people who can do 1 or the OTHER?

OR

Is there a MINIMUM 5 people who can ONLY Bat and a MINIMUM 5 people who can ONLY Bowl?


Going with the former interpretation, the answer is NOT Given.

So going with the latter interpretation:

Since every Wicket is either part of 2 Sets or part of 3 Sets:

The No. of people who are part of EXACTLY 1 Set = (Bat Only - X only) + (Bowl Only - Y only)

and we are told that EACH has to have a MINIMUM 5 people


MIN No. of people who are part of EXACTLY 1 Set = (at least 5 in X) + (at least 5 in Y) = at least 10



Q- what is the MAXIMUM Amount of people who are part of ALL 3 SETS?

Out of 20 Unique people:

We know there are 9 people who are part of EXACTLY 2 Sets

We know AT MINIMUM there must be at least 10 people who are part of EXACTLY 1 Set

and because the 20 people are on the Team, they all must be doing something so there is NO ONE in the Neither Group


The MAXIMUM Number of People who Can Do all 3 Things =

(20 Unique ppl) - (9 ppl part of Exactly 2 Sets) - (at least 10 ppl part of Exactly 1 Set ) =


1 Person

-B-

Stated much better than I stated the misleading part of the question.

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