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Each of three charities in Novel Grove Estates has 8 persons

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Each of three charities in Novel Grove Estates has 8 persons

by BTGmoderatorLU » Sat Jul 07, 2018 3:08 pm

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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 OA is B.

Please, can someone assist me with this PS question? I'm not sure but I think that I can use the Venn Diagram to solve it. Thanks in advance!
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by [email protected] » Sun Jul 08, 2018 6:38 pm
Hi All,

We're told that each of three charities in Novel Grove Estates has 8 persons serving on its board of directors and EXACTLY 4 persons serve on 3 boards each and each pair of charities has 5 persons in common on their boards of directors. We're asked for the number of distinct persons that serve on one or more boards of directors. You will likely find it easiest to use a few drawings and some 'brute force' work to help you to answer this question.

To start, let's focus on the 4 people who are on ALL 3 boards. We'll call those people A, B, C and D...

Charity 1: ABCD _ _ _ _
Charity 2: ABCD _ _ _ _
Charity 3: ABCD _ _ _ _

Next, we need each PAIR of charities to have 5 people in common (the first 4 plus one more); that 5th person CAN'T be on ALL 3 charities though, so we'll need a few additional people to make that happen:

Charity 1: ABCD E F _ _
Charity 2: ABCD E G _ _
Charity 3: ABCD F G _ _

The last two people on each charity have to be 'unique' (meaning that they each appear on ONLY that one charity....

Charity 1: ABCD E F M N
Charity 2: ABCD E G P Q
Charity 3: ABCD F G R S

Total people: ABCDE FGMNP QRS = 13 people

Final Answer: B

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by Shahrukh@mbabreakspace » Wed Jul 11, 2018 6:41 am
There are three types of directors: In One charity, In two Charity and In 3 charity
There are 4 directors in the 3 charities
And 3 directors in 2 charities(Because it says that 5 are common between each pair out of these 4 are in all three but 1 is unique to that pair and there are 3 pairs possible)
So, director in 1 charity are 3*(8-4-2)=6
So, total= 6+3+4= 13
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by GMATGuruNY » Wed Jul 11, 2018 12:42 pm
BTGmoderatorLU wrote: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
This problem involves 3 overlapping sets: Charity A, Charity B, and Charity C.
To organize the data, use a Venn Diagram:
Image

Start in the CENTER and work outwards.

Exactly 4 persons serve on 3 boards each.
Inserting 4 people into the center of the diagram, we get:
Image

Each pair of charities has 5 persons in common.
Thus -- in addition to the 4 people that all 3 charities have in common -- each pair must have 1 additional person in common:
Image

Since each charity must have a total of 8 people, we get:
Image

How many distinct persons serve on one or more boards of directors?
Adding together the values in the Venn Diagram, we get:
2+1+2+1+4+1+2 = 13.

The correct answer is B.
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by Scott@TargetTestPrep » Fri Apr 26, 2019 2:37 pm
BTGmoderatorLU wrote: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

We can use the formula:

n(At least one set) = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C)

Thus, we have:

n(At least one set) = 8 + 8 + 8 - 5 - 5 - 5 + 4 = 24 - 15 + 4 = 13

Answer: B

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