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A club with a total membership of 30 has formed 3 committees

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A club with a total membership of 30 has formed 3 committees  [#permalink]

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New post 11 Dec 2010, 08:32
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A club with a total membership of 30 has formed 3 committees, M, S, and R, which have 8, 12, and 5 members, respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?

A. 5
B. 7
C. 8
D. 10
E. 12
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Re: A club with a total membership of 30  [#permalink]

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New post 11 Dec 2010, 08:50
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ajit257 wrote:
A club with a total membership of 30 has formed 3 committees, M, S, and R, which have 8, 12, and 5 members, respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?
A. 5
B. 7
C. 8
D. 10
E. 12


How to solve these types ?


As "no member of committee M is on either of the other 2 committees" then 30-M=30-8=22 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees

General rule for such kind of problems:
to maximize one quantity, minimize the others;
to minimize one quantity, maximize the others.


So we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S, which has 12 members (if R is subset of S), then total # of members of committee S and committee R would be minimized and equal to 12. Which means that 22-12=10 is the greatest possible number of members in the club who are on none of the committees.

Answer: D.

Also discussed here: a-club-with-a-total-membership-of-104850.html?

Hope it's clear.
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Re: A club with a total membership of 30  [#permalink]

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New post 11 Dec 2010, 09:28
I also got D. Bunuel explained it better than I ever could.
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Re: A club with a total membership of 30  [#permalink]

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New post 24 Jun 2011, 01:38
Got it!! Great question. I don't ever want to miss an overlapping question again.
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Re: A club with a total membership of 30  [#permalink]

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New post 24 Jun 2011, 16:53
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Committee M has 8 members and they cannot be part of any other committee . So that is 8.

We want to get the maximum possible people who are not part of any committee.

=> Committee R and S could have members as follows

R = 5 ,all these 5 could be part of s too
S = 12 .5 already counted . so 7 more distinct people are part of S.

i.e 30-8-5-7 = 10

Answer is D.
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Re: A club with a total membership of 30  [#permalink]

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New post 25 Jun 2011, 07:49
Total club members = 30

no members of M is in any other club. Hence, exclusive members in M = 8
Now, for max. of members who are not part of any group, there might be 5 members in R who all are in group S.

Hence, total number of members in group = members in M + members in S
12+8 = 20.

hence, number of members who are not in any group= 30-20 =10.
IMO D
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Re: A club with a total membership of 30  [#permalink]

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New post 25 Jun 2011, 10:04
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Find the image below.Hope you like it.
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ans.png [ 14.83 KiB | Viewed 10740 times ]


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Re: A club with a total membership of 30  [#permalink]

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New post 10 Sep 2011, 06:03
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Attachment:
Screen Shot 2011-09-10 at 4.01.18 PM.png
Screen Shot 2011-09-10 at 4.01.18 PM.png [ 29.3 KiB | Viewed 10345 times ]


Here's a Venn Diagram.
total-m-r-s= 30-8-7-5=5
Also 5 already are not there in any committee.

Total=5+5=10

OA D
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Re: A club with a total membership of 30  [#permalink]

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New post 08 Mar 2012, 03:15
8+12-both+5-neither=30
neither -both=5

to maximize neither we need to minimize both. if both=5, then neither =10
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Re: A club with a total membership of 30  [#permalink]

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New post 31 May 2013, 13:47
Total number of people = 30
Number of people on m = 8
So pool of people left = 22

To maximize people on no committees assume all members of R are also part of S

So people on no committees = 22 - 12= 10
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Re: A club with a total membership of 30 has formed 3 committees  [#permalink]

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New post 23 Jul 2014, 23:23
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Refer Venn Diagram

Setting up the equation:

8 + 5 + 12 - x + y = 30

y - x = 5

y = x + 5

To have maximum value of y (Shaded in green), x has to be maximum

x is the common area (shaded in pink) which can have maximum value of 5

y = 5 + 5 = 10

Answer = D
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Re: A club with a total membership of 30 has formed 3 committees  [#permalink]

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New post 04 Jan 2019, 12:34
Hi Gladiator59

Just a prompt question the problem asks
"what is the greatest possible number of members in the club who are on none of the committees" I know how to get the answer (by merging R to S) thus , in this way only R will be equal to 0 , however even in this way those 5 who were merged with S are still part of committee R right? I mean that they now belong to both R and S but how can we deduce that those 5 have seized to be R members and are just S members after the merging.
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Re: A club with a total membership of 30 has formed 3 committees  [#permalink]

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New post 04 Jan 2019, 13:12
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To answer your question - Yes. They would still continue to be members of R but now they would also be members of S and no "new members from the 30 total" would be needed to populate R. This, in turn, would maximize the no. of people who are in neither MSR.

So M is unique and has 8 people who are only in M. Now with the maximizing step, we have 12 members in S ( among these 12 there are 5 who are also in R). So finally only 12 + 8 = 20 are in either of the committees. Hence 10 are such that they are in neither.

Hope it is clear. If you can imagine the solution this question can be solved in a matter of seconds.

UNSTOPPABLE12 wrote:
Hi Gladiator59

Just a prompt question the problem asks
"what is the greatest possible number of members in the club who are on none of the committees" I know how to get the answer (by merging R to S) thus , in this way only R will be equal to 0 , however even in this way those 5 who were merged with S are still part of committee R right? I mean that they now belong to both R and S but how can we deduce that those 5 have seized to be R members and are just S members after the merging.

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Re: A club with a total membership of 30 has formed 3 committees   [#permalink] 04 Jan 2019, 13:12
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