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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] New post 11 Dec 2010, 07:32
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60% (02:03) correct 40% (01:29) wrong based on 141 sessions
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
[Reveal] Spoiler: OA

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Re: A club with a total membership of 30 [#permalink] New post 11 Dec 2010, 07:50
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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] New post 11 Dec 2010, 08:28
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] New post 24 Jun 2011, 00:38
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] New post 24 Jun 2011, 15:53
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] New post 25 Jun 2011, 06:49
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] New post 25 Jun 2011, 09:04
Find the image below.Hope you like it.
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Re: A club with a total membership of 30 [#permalink] New post 10 Sep 2011, 05:03
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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] New post 08 Mar 2012, 02:15
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] New post 29 May 2013, 06:36
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Re: A club with a total membership of 30 [#permalink] New post 31 May 2013, 12:47
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] New post 15 Jul 2014, 05:03
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Re: A club with a total membership of 30 has formed 3 committees [#permalink] New post 23 Jul 2014, 22:23
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] 23 Jul 2014, 22:23
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