Can u please help me with this question? I don't quite get the picture. :/

Question

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

Answer 1

The inclusion exclusion tells you that n(a U B U C) = n(A) + n(B) + n(C) - n(AB) - n(AC) - n(BC) + n(ABC) =

3 * 8 - 3 * 5 + 4 = 13

Note how we could reason this in a different way.

Take the 8 people in Club 1. 5 are in club 2 and 5 are in club 3, so you can remove these 10.

Then, consider the remaining 3 in club 2. Since 4 of the ones already removed were among the ones removed from club 3 (they were in all 3 clubs), and there are 5 in clubs 2 and 3, we remove 1 additional one from club 3. Thus, we are left with 3 in club 2 and 2 in club 3.

8 + 3 + 2 = 13