Set Theory

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Uva@90: Master | Next Rank: 500 Posts; Posts: 490; Joined: Thu Jul 04, 2013 7:30 am; Location: Chennai, India; Thanked: 83 times; Followed by:5 members

Set Theory

by Uva@90 » Sat Aug 08, 2015 3:46 am

Each of three charities in Sn Royalty Estates has 16 persons servings on its board of directors. If exactly 8 persons serve on 3 board each and each pair of charities has 10 persons in common on their board of directors, then how many distinct persons serve on one or more board of directors ?

A) 16
B) 26
C) 32
D) 48
E) 54

OA B

Thanks in advance.

Regards,
Uva.

Known is a drop Unknown is an Ocean

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Source: Beat The GMAT — Problem Solving | Sat Aug 08, 2015 3:46 am

MartyMurray: Legendary Member; Posts: 2135; Joined: Mon Feb 03, 2014 9:26 am; Location: https://martymurraycoaching.com/; Thanked: 955 times; Followed by:140 members; GMAT Score:800

by MartyMurray » Sat Aug 08, 2015 4:50 am

This one can be answered via drawing a Venn diagram or by using some formula.

There is a formula that's perfect for this. So, even though I rarely use formulas, I am going to use that.

Just be sure to understand why the formula works, and that it applies to this situation but may not work in a slightly different situation. There are other formulas that work better in other situations, and you can also come up with your own formula if you understand how this all works.

When we have three groups as we do in this situation, and we know the overlaps of the groups, but are not given exactly how many are in exactly two groups, we can use the following formula.

Total = A + B + C - (sum of two group overlaps) + (1 x the three group overlap)

That works because we eliminate all the overlaps between pairs of groups and then add back the number who serve on all three, because via subtracting the pairs we subtracted the number who sit on all three three times and we need it to show up.

16 + 16 + 16 - (10 + 10 + 10) + 8 = 26 People

Choose B.

Marty Murray
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Uva@90: Master | Next Rank: 500 Posts; Posts: 490; Joined: Thu Jul 04, 2013 7:30 am; Location: Chennai, India; Thanked: 83 times; Followed by:5 members

by Uva@90 » Sat Aug 08, 2015 5:10 am

Marty Murray wrote:This one can be answered via drawing a Venn diagram or by using some formula.

There is a formula that's perfect for this. So, even though I rarely use formulas, I am going to use that.

Just be sure to understand why the formula works, and that it applies to this situation but may not work in a slightly different situation. There are other formulas that work better in other situations, and you can also come up with your own formula if you understand how this all works.

When we have three groups as we do in this situation, and we know the overlaps of the groups, but are not given exactly how many are in exactly two groups, we can use the following formula.

Total = A + B + C - (sum of two group overlaps) + (1 x the three group overlap)

That works because we eliminate all the overlaps between pairs of groups and then add back the number who serve on all three, because via subtracting the pairs we subtracted the number who sit on all three three times and we need it to show up.

16 + 16 + 16 - (10 + 10 + 10) + 8 = 26 People

Choose B.

Thanks Muray,

I got confused with the below line,

"exactly 8 persons serve on 3 board each" so here 8 is the value for three charities together.
I understood as 8 server on each charity alone... Thats my mistake..

Thanks for clarifying.

Regards,
Uva.

Known is a drop Unknown is an Ocean

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by [email protected] » Sat Aug 08, 2015 10:38 am

Hi Uva@90,

There's also a 'visual way' to keep track of the 3 groups and the people who are in one, two or three of the groups:

We're told that there are 3 charities with 16 people on each. EXACTLY 8 serve on all 3....Let's call those 8 people: ABCDEFGH

1) ABCDE FGH
2) ABCDE FGH
3) ABCDE FGH

Next, we're told that each pair of charities has 10 people IN COMMON. Right now, each pair of charities has 8 people in common. We now have to add two more people to each PAIR of charities...

1) ABCDE FGH | JK LM -- |
2) ABCDE FGH | JK -- NO |
3) ABCDE FGH | -- LM NO |

Now each pair of charities has 10 people in common. So far, that is a total of 8 + 6 = 14 people.

Any additional people have to be UNIQUE to their respective charities. Since each charity has a TOTAL of 16 people on it AND we've already accounted for (8+4) people per charity, there must be 4 UNIQUE ADDITIONAL people per charity...(3 charities)(4 unique people each) = 12 additional people....

14 + 12 = 26 people.

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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MartyMurray: Legendary Member; Posts: 2135; Joined: Mon Feb 03, 2014 9:26 am; Location: https://martymurraycoaching.com/; Thanked: 955 times; Followed by:140 members; GMAT Score:800

by MartyMurray » Tue Aug 11, 2015 8:05 am

Rich, that is so cool. Just what I was craving.

Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.

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by GMATGuruNY » Tue Aug 11, 2015 8:34 am

Uva@90 wrote:Each of three charities in Sn Royalty Estates has 16 persons servings on its board of directors. If exactly 8 persons serve on 3 board each and each pair of charities has 10 persons in common on their board of directors, then how many distinct persons serve on one or more board of directors ?

A) 16
B) 26
C) 32
D) 48
E) 54

Let the 3 charities be A, B and C.
Since each charity's board consists of 16 people, use the following VENN DIAGRAM:

Now work from the INSIDE OUT.

Exactly 8 persons serve on 3 boards each:

Each pair of charities has 10 persons in common:

Since each charity must have a total of 16 people, we get:

How many distinct persons serve?
Adding together the values in the Venn Diagram, we get:
4+2+4+2+8+2+4 = 26.

The correct answer is B.

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