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by GMATSUCKER » Mon Mar 01, 2010 12:09 pm
Q: Each of three charities in Novel Grove Estates has 8 persons serving on its board of directors.If exactly 4 persons serve on 3 boards each and each pair of charities has 5 persons in common on their boards of directors ,then how many distinct persons serve on one or more boards of directors?

a)8

b)13

c)16

d)24

e)27
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by Stuart@KaplanGMAT » Mon Mar 01, 2010 1:45 pm
GMATSUCKER wrote:Q: Each of three charities in Novel Grove Estates has 8 persons serving on its board of directors.If exactly 4 persons serve on 3 boards each and each pair of charities has 5 persons in common on their boards of directors ,then how many distinct persons serve on one or more boards of directors?

a)8

b)13

c)16

d)24

e)27
The easiest way to solve this is via brute force. Let's work it out:

3 charities, 8 slots each. Therefore, we have 24 slots to fill.

4 people sit on all 3 boards - let's start to fill slots:

Ch 1: ABCD
Ch 2: ABCD
Ch 3: ABCD

Each pair has exactly 5 persons in common. Well, 4 of those are ABCD, so each pair of charities has to have one more person in common. To make that work, we need exactly 1 person on each pair of charities. So, we now have:

Ch 1: ABCDEF
Ch 2: ABCDEG
Ch 3: ABCDFG

(ABCDE are on 1/2, ABCDF are on 1/3, ABCDG are on 2/3, fulfilling the requirement). So far we've used up A through G.

We have 2 slots left to fill on each board. Since we're not allowed any more overlap, we need 6 individuals for those 6 slots:

Ch 1: ABCDEFHI
Ch 2: ABCDEGJK
Ch 3: ABCDFGLM

We've used up A through M: that's 13 letters. Choose (b)!
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by goalevan » Mon Mar 01, 2010 6:46 pm
I find it easier conceptually to quickly sketch a Venn diagram, and start with the problem with the middle region.

The boards have four people in common:

In the center region shared by all four clubs, mark a 4.

Each pair has five people in common:

In the regions shared by two clubs, mark a 1. (5-4=1)

Each club has eight people in total:

In the regions exclusive to one club, mark a 2. (8-4-1-1=2)


Counting all regions on the diagram, you will get 4 + 1 + 1 + 1 + 2 + 2 + 2 = 13
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by GMATSUCKER » Tue Mar 02, 2010 11:18 am
Ans.B . Source :Kaplan Advanced. Now the answer suggested in that book is quite lengthy and time consuming. So is there any other fast and shortcut approach to this problem.
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by Stuart@KaplanGMAT » Tue Mar 02, 2010 12:03 pm
GMATSUCKER wrote:Ans.B . Source :Kaplan Advanced. Now the answer suggested in that book is quite lengthy and time consuming. So is there any other fast and shortcut approach to this problem.
You mean other than the two posted here?

Each of the approaches in this thread are doable in well under 2 minutes - they key is to think of them quickly. The implementation is pretty fast, it's coming up with the approach that often takes too long.

So, as you study, keep looking for multiple approaches - that way you're more likely to find the quickest one when you need it: on test day.
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by stufigol » Tue Mar 02, 2010 10:25 pm
what s the 1 u dewleted in 8-4-1-1
i mean the last one
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by Stuart@KaplanGMAT » Wed Mar 03, 2010 2:35 pm
stufigol wrote:what s the 1 u dewleted in 8-4-1-1
i mean the last one
Looking at group 1, for example:

8 people total
4 people in all 3 groups
1 person doubles up with group 2
1 person doubles up with group 3

So, each "1" represents one of the doubles for that group.

Looking at the "group 1" circle in our venn diagram, it has 8 total members and 4 components:

Group 1 only = ?
Group 1/2 = 1
Group 1/3 = 1
Group 1/2/3 = 4

8 - 4 - 1 - 1 = 2, which means that the "group 1 only" part of the venn diagram has 2 people in it.
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