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# Each of three charities in Novel Grove Estates has 8 persons

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Intern
Joined: 19 Dec 2010
Posts: 28
Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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Updated on: 13 Aug 2014, 06:56
17
00:00

Difficulty:

55% (hard)

Question Stats:

65% (02:23) correct 35% (02:41) wrong based on 197 sessions

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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

Attachments

Overlapping Problem.jpg [ 12.62 KiB | Viewed 4081 times ]

Problem.pdf [24.45 KiB]

Originally posted by m990540 on 30 Jan 2011, 18:26.
Last edited by Bunuel on 13 Aug 2014, 06:56, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Location: Pune, India
Re: Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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31 Jan 2011, 20:26
4
4
m990540 wrote:
Sorry, the answer is 13. I know that this question can be found in the forum, but I'd like to know if there's a way (such as using the standard overlapping set equation) to attack the problem other than by drawing out each of the three groups!

Thanks so much!

You can use the formula for 3 sets:
n(AUBUC) = n(A) + n(B) + n (C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C)

n(AUBUC) = 8 + 8 + 8 - 5 - 5 -5 + 4 = 13

Though, why would you not try using the Venn diagram for 3 overlapping sets. This question is relatively straight forward so the formula is no problem, but with the Venn diagram, it doesn't matter how much you twist the question, the answer is still apparent....
I am attaching it in case you would like to give it a look...
Attachment:

Ques2.jpg [ 10.5 KiB | Viewed 4024 times ]

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Karishma
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##### General Discussion
Intern
Joined: 19 Dec 2010
Posts: 28
Re: Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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30 Jan 2011, 18:51
Sorry, the answer is 13. I know that this question can be found in the forum, but I'd like to know if there's a way (such as using the standard overlapping set equation) to attack the problem other than by drawing out each of the three groups!

Thanks so much!
Math Expert
Joined: 02 Sep 2009
Posts: 52906
Re: Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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01 Feb 2011, 02:08
m990540 wrote:
Hello,

Here's an overlapping set question that I came across and obviously got wrong.

Thanks in advance to anyone who can help me sort this one out!

Check this link for different formulas on 3 overlapping sets (theory and examples): formulae-for-3-overlapping-sets-69014.html#p729340
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Re: Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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01 Feb 2011, 05:50
Wow, thanks for the great explanations and link!
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Re: Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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05 Aug 2014, 20:51
1
Distinct persons means addition of all colours (As per diagram below)

= 2+1+4+1+2+1+2

= 13

Attachments

Ques2.jpg [ 16.69 KiB | Viewed 2684 times ]

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Posts: 118
Re: Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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14 Aug 2015, 00:52
m990540 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

A intersec B intersec C = 4
A intersec B but not C = 1
B intersec C but not A = 1
C intersec A but not B = 1
A only is 2; B only is 2; C only is 2
Therefore, total 13 members serve on one or more boards of directors.
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Joined: 22 Sep 2018
Posts: 241
Re: Each of three charities in Novel Grove Estates has 8 persons  [#permalink]

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22 Jan 2019, 11:51
m990540 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

Interesting question. Here's my reasoning:

# of people who are on the board of exactly two charities: 5-4 = 1 -> 3 charities in the estates so 1*3 = 3 people total on 2 boards.

3 charities with overlap (double counting exists here) have a total of 24 people (8 *3).

We are double counting the people on exactly 2 boards and triple counting people on exactly 3 boards. 24 - (3*2) - (4*3) = 6. 6 people are on exactly one board.

Add it all up (6 + 3 + 4) and we get the answer 13
Re: Each of three charities in Novel Grove Estates has 8 persons   [#permalink] 22 Jan 2019, 11:51
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