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Greatest possible number

by mgmt_gmat » Fri Feb 12, 2010 1:05 am
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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by sanju09 » Fri Feb 12, 2010 5:55 am
mgmt_gmat 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
If no member of committee M is on either of the other 2 committees, then two cases arise:

1. The other 2 committees, S and R have member(s) in common.

2. The other 2 committees, S and R do not have a member in common.

For maximizing the number of members in the club who are on none of the committees, we will have to take the case 1 at its best in minimizing the net number of members in the other 2 committees, S and R.

When M is already away with 8 members, we are left with 22 members to discuss. If R is contained in S, n (S ∪ R) = n (S) = 12.

Hence, at most [spoiler]10 (subtracted 17 in place of 12 from 22, initially, hence edited)[/spoiler] members in the club could be on none of the committees.

[spoiler]D[/spoiler]
Last edited by sanju09 on Fri Feb 12, 2010 6:07 am, edited 1 time in total.
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by ajith » Fri Feb 12, 2010 6:02 am
mgmt_gmat 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
No member of M can be a member of other committees and M has 8 members.

If we exclude these 8 members, we still have 22 left

Out of these 22 - 12 are members of S and 5 are members of R

The maximum no of people are not in any committee when all the five members of R are the members of S as well.

The maximum no of people are not in any committee in that case = [spoiler]22 - 12 = 10, D[/spoiler]
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by ajith » Fri Feb 12, 2010 6:04 am
sanju09 wrote: When M is already away with 8 members, we are left with 22 members to discuss. If R is contained in S, n (S ∪ R) = n (S) = 12.

Hence, at most 5 members in the club could be on none of the committees.

[spoiler]A[/spoiler]
I disagree only with the bold part - How did u get the number 5, isn't it 22-12 = 10?
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