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# Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to

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Manager
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Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to [#permalink]

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18 May 2010, 15:52
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Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to Z. If 2 belong to all three organizations and 3 belong to exactly 2 organizations, how many belong to none of these organizations?

Sorry I don't have OA.
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23 May 2010, 11:36
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The answer should be 10. This is how I got to it,

Total - 20
Group X - 5
Group Y - 7
Group Z - 9

Common Between 2 Groups - 3 (Lets take this as X+Y)
Common Between 3 Groups - 2 (Lets take this as X+Y+Z)
Neither - N

Total - N = (X+Y+Z) - 2(common b/w 2 groups) - 2(common b/w 3 groups)
20 - N = 20 - (4 + 6)
20 - N = 10 & N = 10

Though you might want to check on the figures because 5,7 & 9 add upto 21 not 20. So if the total is 20 then 10 are part of neither.

Roy
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24 May 2010, 03:00
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pardeepattri wrote:
Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to Z. If 2 belong to all three organizations and 3 belong to exactly 2 organizations, how many belong to none of these organizations?

Sorry I don't have OA.

Above solution is not right.

The formula would be:

ALL = 20 = X+Y+Z - {Members of Exactly 2} - 2*{Members of All 3} + None --> 20=5+7+9-3-2*2+N --> N=6.

We subtract {Members of Exactly 2} once as when we add X+Y+Z, this intersection (exactly 2 - segments 1, 2 and 3 on the diagram below) is counted twice thus one should be subtracted.

We subtract {Members of All 3} twice as when we add X+Y+Z, this intersection (all 3 - segment 4 in the diagram below) is counted thrice, thus two should be subtracted.
Attachment:

Union_3sets.gif [ 11.63 KiB | Viewed 8021 times ]

For more check ADVANCED OVERLAPPING SETS PROBLEMS
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24 May 2010, 06:00

my quant is bit rusty and venn diagrams is something i really want to improve at, thank Quant Forum Moderator

Kudos to you!

Roy
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26 Jul 2010, 04:12
thanks for sharing this question
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25 Jun 2011, 09:15
Total = (Neither) + x + y + z - (Exactly 2) - 2*(All 3)

20 = (N) + 21 - (3) - (4)

20 = 14 + N

Neither = 6

Good question this -- helps if one knows the formula.
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25 Jun 2011, 17:20
total = X+Y+Z-2(all three) -(sum of two overlaps) + neither

20 = 5+ 7+9 - 2(2)-3+neither
=>neither = 6
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Re: Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to [#permalink]

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07 Jun 2014, 00:10
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Re: Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to [#permalink]

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28 Jun 2015, 03:06
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to [#permalink]

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20 Jan 2016, 20:36
safaria25 wrote:
Bunuel wrote:
pardeepattri wrote:
Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to Z. If 2 belong to all three organizations and 3 belong to exactly 2 organizations, how many belong to none of these organizations?

Sorry I don't have OA.

Above solution is not right.

The formula would be:

ALL = 20 = X+Y+Z - {Members of Exactly 2} - 2*{Members of All 3} + None --> 20=5+7+9-3-2*2+N --> N=6.

We subtract {Members of Exactly 2} once as when we add X+Y+Z, this intersection (exactly 2 - segments 1, 2 and 3 on the diagram below) is counted twice thus one should be subtracted.

We subtract {Members of All 3} twice as when we add X+Y+Z, this intersection (all 3 - segment 4 in the diagram below) is counted thrice, thus two should be subtracted.

Attachment:
Union_3sets.gif

For more check ADVANCED OVERLAPPING SETS PROBLEMS

Hi Bunuel,

I'm not sure when to use the first or second formulas (GMAT CLUB Quant Book) for these Venn Diagram questions, can you please help?

Thanks!

The very post you quote has a link which explains it. Here it is again: ADVANCED OVERLAPPING SETS PROBLEMS
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Of 20 Adults, 5 belong to X, 7 belong to Y, and 9 belong to   [#permalink] 20 Jan 2016, 20:36
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