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What is the best approach here? I tryed the venn diagram, [#permalink]
26 Jul 2003, 03:21

What is the best approach here? I tryed the venn diagram, but it did not help.

Q.
Each of three charities in East Side 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?

if each pair of charities has 5 person in common, then this number includes those who serve in all the charities. Therefore, the number of people who serve in EXACTLY 2 charities is 1.

I wrongly assumed the meaning of "each pair of charities has five persons in common" as 5 persons serving in each pair only and was stuck because the total number appeared negative.

imagine three overlapping circles. you need to find an area of this construction. BUT!!! you cannot simply add areas of the given circles. some areas overlap twice and one thrice. you have to eliminate these pieces in order not to count them many times.

So, we have to eliminate 'double' regions, and 'triple' region. In doing so, we 'clear' our construction from any overlapping.

AREA=A+B+C-exactly AB-exactly AC-exactly BC-2*ABC

The problem is much complicated when it comes to deal with four and more overlapping sets.

Last edited by stolyar on 26 Jul 2003, 12:12, edited 1 time in total.

You should clearly understand what double and triple regions are. We have three double regions and one triple region.

Three double regions are counted twice. We need to count them only once; therefore, we need to subtract them from the sum.

The triple region is counted thrice. We need to subtract two times it from the sum.

To understand all this matter better, play with three paper circles. Overlap them, and cut them to have only one layer of paper throughout the whole construction.