Kaplan question-Confused on how to approach !!!

[karta]

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 director ?
A-8
B-13
C-16
D-24
E-27

[raja501]

Let the three charities be X, Y, Z. We are told that 8 people are in each charity board. So, let's identify the board positions as
X1, X2, X3, X4, X5, X6, X7, X8
Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8
Z1, Z2, Z3, Z4, Z5, Z6, Z7, Z8

Exactly 4 people serve on charity X, Y and Z.
So, X1, X2, X3, X4 = Y1, Y2, Y3, Y4, = Z1, Z2, Z3, Z4.

Each pair of charities has 5 persons in common.
Since each pair of charities already has 4 people in common, we need to add the fifth. i.e, one more person in common between two charities.

X5 = Y5, Y6 = Z5, X6 = Z6.

So, the total number of board members at this point is 7 occupying 6 seats in each board.

There are still 2 board positions remaining for each charity and each board position is occupied by a unique person. So, that makes a total of 6 people occupying the remaining 6 board positions (2 for X, 2 for Y and 2 for Z)

So, the total number is 7+6 = 13. Option B.

[ascnap]

I initially thought the answer was 8. But raja's explanation above made more sense. Another way to look at this would be through Venn diagram.
If you draw the Venn diagram, you can see that 4 persons (serve on all three - and hence would occupy the intersection of all 3 circles). Then among 2 circles (charities) there are 5 persons in common. So 4 + 1 (between any 2 charities). So now the total number of distinct persons would be 2*3 + 1*3 + 4 = 13.
2*3 (dinstinct members in each charity)
4 (amongst all 3 charities)
and finally 1*3 (5-4) common between 2 charities.

[krusta80]

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 director ?
A-8
B-13
C-16
D-24
E-27

This is all about putting the problem into logical terms and then applying what we know.

1. "Each of three charities in Novel Grove Estates has 8 persons serving on its Board of Directors."

Let A be the number of board directors for the first charity
Let B be the number of board directors for the second charity
Let C be the number of board directors for the third charity
A = B = C = 8


2. "exactly 4 persons serve on 3 boards each"

(A and B and C) = 4


3. "each pair of charities has 5 persons in common on their boards of directors"

(A and B) = (A and C) = (B and C) = 5


4. "how many distinct persons serve on one or more boards of director ?"

(A or B or C) = ?


Now we solve!

(A or B or C) = A + B + C - (A and B) - (A and C) - (B and C) + (A and B and C)

(A or B or C) = 8 + 8 + 8 - 5 - 5 - 5 + 4 = 13