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# M04-19

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Math Expert
Joined: 02 Sep 2009
Posts: 54371

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16 Sep 2014, 00:23
1
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35% (medium)

Question Stats:

70% (01:10) correct 30% (01:34) wrong based on 193 sessions

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6 students in a group study different languages as specified:

Russian: 4

Ukrainian: 3

Hebrew: 2

Each student studies at least 1 language. If it is also known that exactly 3 students study exactly 2 languages, how many students are studying all three languages?

A. 0
B. 1
C. 2
D. 3
E. 4

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Math Expert
Joined: 02 Sep 2009
Posts: 54371

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16 Sep 2014, 00:23
1
3
Official Solution:

6 students in a group study different languages as specified:

Russian: 4

Ukrainian: 3

Hebrew: 2

Each student studies at least 1 language. If it is also known that exactly 3 students study exactly 2 languages, how many students are studying all three languages?

A. 0
B. 1
C. 2
D. 3
E. 4

We have a total of 9 classes taken by 6 students. If 3 of them study 2 languages (which makes $$3*2=6$$ classes), we have $$9-6=3$$ classes and $$6-3=3$$ students left. Therefore, as students are studying at least one language, nobody is taking all 3.

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Joined: 25 Feb 2014
Posts: 45
GMAT 1: 730 Q50 V39
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02 Nov 2014, 15:07
Bunuel wrote:
Official Solution:

6 students in a group study different languages as specified:

Russian: 4

Ukrainian: 3

Hebrew: 2

Each student studies at least 1 language. If it is also known that exactly 3 students study exactly 2 languages, how many students are studying all three languages?

A. 0
B. 1
C. 2
D. 3
E. 4

We have a total of 9 classes taken by 6 students. If 3 of them study 2 languages (which makes $$3*2=6$$ classes), we have $$9-6=3$$ classes and $$6-3=3$$ students left. Therefore, as students are studying at least one language, nobody is taking all 3.

Did nt understand the explanation.
I agree that 6 classes are taken by 3 who study 2 languages.
3 classes are left. Now 3 students can be taking all 3 classes or 3 students can be taking individual 3 subject classes. How does 3 students taking 3 classes can be ruled out as in both these cases, each student is studying atleast 1 language.
Math Expert
Joined: 02 Sep 2009
Posts: 54371

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03 Nov 2014, 03:28
mayankpant wrote:
Bunuel wrote:
Official Solution:

6 students in a group study different languages as specified:

Russian: 4

Ukrainian: 3

Hebrew: 2

Each student studies at least 1 language. If it is also known that exactly 3 students study exactly 2 languages, how many students are studying all three languages?

A. 0
B. 1
C. 2
D. 3
E. 4

We have a total of 9 classes taken by 6 students. If 3 of them study 2 languages (which makes $$3*2=6$$ classes), we have $$9-6=3$$ classes and $$6-3=3$$ students left. Therefore, as students are studying at least one language, nobody is taking all 3.

Did nt understand the explanation.
I agree that 6 classes are taken by 3 who study 2 languages.
3 classes are left. Now 3 students can be taking all 3 classes or 3 students can be taking individual 3 subject classes. How does 3 students taking 3 classes can be ruled out as in both these cases, each student is studying atleast 1 language.

Tip: try to draw Venn diagram and see if it's possible.
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Joined: 08 Apr 2013
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13 Mar 2015, 14:54
Can someone please solve this using a Venn diagram
Manager
Joined: 05 Feb 2015
Posts: 51
Concentration: Finance, Entrepreneurship
Schools: ISB '16, IIMA , IIMB, IIMC
WE: Information Technology (Health Care)

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13 Mar 2015, 21:08
Hi..

I could not find an option to insert an image so I have attached it.
Consider these 3 circles to be one of Russian, Ukranian and Hebrew each.

We have, R+U+H=9
The shaded part in red should be 3*2=6
This leaves us with 9-6=3
3 could not be the green part as each student studies atleast 1 language. Because if that were the case, then we have only 4 students.
So, the answer should be 0. No one studies all the 3 languages.
>> !!!

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Intern
Joined: 07 May 2015
Posts: 16
Location: India
GPA: 3.5
WE: Information Technology (Computer Software)

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16 Jul 2015, 03:46
Hi Naina1.. Can you please guide me on this part "The shaded part in red should be 3*2=6". As i am not able to co relate it in problem. Thanks
Current Student
Joined: 23 Jan 2015
Posts: 3

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16 Jul 2015, 07:38
Is it correct to then conclude the following:

2 students studying only R
1 student studying only U
0 students studying only H

1 student studying both R+H
1 student studying both R+U
1 student studying both U+H

0 students studying all three

2+1+0+1+1+1=6

Is this correct?

Is there a formulaic approach using the overlapping sets formulas to see this?
Intern
Joined: 11 Jun 2015
Posts: 2
Concentration: Strategy, Technology
GMAT Date: 09-19-2015
GPA: 3.8

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16 Jul 2015, 19:13
1
6 Students-> A B C D E F
* * * * * * -> 6 enrollment - each student studies at least one subject
* * * -> 3 enrollment - exactly 3 students study exactly 2 languages
Total Enrollment = 6 + 3 = 9
As Given Total Enrollment= 4(R) + 3(U) + 2(H) =9 , so no student studies all 3.
Manager
Joined: 23 Jun 2009
Posts: 178
Location: Brazil
GMAT 1: 470 Q30 V20
GMAT 2: 620 Q42 V33

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05 Aug 2016, 07:16
1
msauers wrote:
Is it correct to then conclude the following:

2 students studying only R
1 student studying only U
0 students studying only H

1 student studying both R+H
1 student studying both R+U
1 student studying both U+H

0 students studying all three

2+1+0+1+1+1=6

Is this correct?

Is there a formulaic approach using the overlapping sets formulas to see this?

I hope it helps
>> !!!

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Intern
Joined: 26 Jul 2016
Posts: 3

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23 Dec 2016, 01:57
4
1
please refer the file for venn diagram approach..... i also tried to derive a generalized formula
>> !!!

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Intern
Joined: 24 Oct 2016
Posts: 6
GPA: 3
WE: Engineering (Manufacturing)

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10 Mar 2017, 14:47
This is an interesting / rare overlapping sets problem that involves triples.
A general formula for overlapping sets with triples is: A + B + C - Doubles - 2(Triples) = Total
In this instance we are given everything except the triples, which when we "plug n' chug" will give you 0 = 2T ~ 0/1 = 0
Intern
Joined: 19 Dec 2016
Posts: 5

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06 Jul 2017, 04:04
The question is pretty simple.

Given Data: 6 students
3 languages.
Each student studies at least 1 language
exactly 3 students study exactly 2 languages

Also we are given students in each language as follows
Russian: 4
Ukrainian: 3
Hebrew: 2 - which totals up to 9

Solution.
So we need to fill slots.
1. Each student studies at least 1 language and we have six students - 6 slots.

2. exactly 3 students study exactly 2 languages - (3 more slots) (since the slot for their first language is already taken in 1.)

9 slots already taken so no student can take 3 languages.
Re: M04-19   [#permalink] 06 Jul 2017, 04:04
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# M04-19

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