M04-19

Question

In a study group of six students, each student is studying at least one of the following languages: Russian, Ukrainian, or Hebrew. The distribution of language studies is as follows:
   
• Russian is studied by four students.
 
• Ukrainian is studied by three students.
 
• Hebrew is studied by two students.
   
If three students from the group are studying exactly two languages, how many students study all three languages?

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

rohitpoonia · Accepted Answer

please refer the file for venn diagram approach..... i also tried to derive a generalized formula

Bunuel · Answer

Official Solution:

In a study group of six students, each student is studying at least one of the following languages: Russian, Ukrainian, or Hebrew. The distribution of language studies is as follows:
   
• Russian is studied by four students.
 
• Ukrainian is studied by three students.
 
• Hebrew is studied by two students.
   
If three students from the group are studying exactly two languages, how many students study all three languages?

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

The formula for calculating the number of students in three overlapping sets is as follows:
 
(Total number of students) = (Students studying Russian) + (Students studying Ukrainian) + (Students studying Hebrew) - (Students studying exactly two languages) - 2*(Students studying all three languages) + (Students studying none of the languages).
 
When we initially add up the students studying Russian, Ukrainian, and Hebrew, the counts have overlaps. For instance, students studying exactly two languages end up being counted twice since they fall into two language groups. Similarly, students studying all three languages get counted three times, once per language group. To correct this, we adjust the total by subtracting the count of students studying exactly two languages once and the count of students studying all three languages twice. This adjustment ensures that each student, regardless of the number of languages they're studying, is only counted once.
 
For more information and examples, check out this link:  ADVANCED OVERLAPPING SETS PROBLEMS 
 
Applying the formula to our case, we have:  6=4+3+2-3-2*x+0 ;
 
Solving gives  x = 0 .
 
Thus, we find that no students are studying all three languages.

Answer: A

Naina1 · Answer

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.

matts55 · Answer

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?

felippemed · Answer

I hope it helps

ocelot22 · Answer

High quality question Bunnell. There is a formula for OL sets, however, the formula is very easy to make a mistake on (and GMAC knows this). For this reason, I don't recommend memorizing the formula. Instead draw a Venn diagram. Place a dot on every section that is included in A. Then place a dot on every section included in B and then every section included in C. If there was no overlap every section should have one dot. What we see is that the 2 OL sections have on extra dot, and the 3 OL section has extra dots. From here we would know that the 2 OL sections should be subtracted out once, and that the 3 OL section should be subtracted out twice. From here it is easy to obtain the correct formula A + B + C - (2 OL's) - 2*(3 OL) = Tot. So 4 + 3 + 2 - 3 - 2*(3 OL) = 6 and -2*(3 OL) = 0, so 3 OL = 0 and OA is A

Attachments

3 OL SETS.png [ 11.74 KiB | Viewed 16423 times ]

AndrewN · Answer

An  alternative approach --This one might prove useful to anyone considering taking the in-home version of the exam, the one without a physical noteboard to write on. I, for one, would not want to be drawing digital Venn diagrams or even scribbling out digital algebraic expressions with multiple variables. Consider a mental  keep-away method  instead, aiming to preserve the slot in question within the context of the given information. If  exactly  3 students study  exactly  2 languages, then both Hebrew slots cannot be filled in this manner, since that would exhaust the 2 and leave no one able to fill another slot in Hebrew. It makes sense to try to fit these three students into the category with the most students, Russian. Consider:

Student 1: Russian + Ukrainian
Student 2: Russian + Ukrainian
Student 3: Russian + Hebrew (since we cannot take up the third Ukrainian slot)

That is 6 slots filled. With 3 students to go, we have just 3 more slots in which to put them. Game over. If any one of these students could take 3 classes, it would mean the other 2 students would have nothing left to take, since the overall enrollment stands at 9.

The question is easily solved in a minute or less using nothing more than simple logic.

- Andrew

Stanindaw · Answer

Bunuel  pls tell the split for venn diagram. Why cant the 2 students of hubrew be a part of exact 2? Only Hebrewcan be 0 so why is that wrong

Bunuel · Answer

Check the diagram below: https://gmatclub.com/forum/download/file.php?id=68287 Given: Total = 6; Russian = 4 ; Ukrainian = 3 ; Hebrew = 2 . Each student studies at least 1 language. None = 0 Exactly 3 students study exactly 2 languages: D + E + F = 3. Question: G = ? Total = Russian + Ukrainian + Hebrew + None - (Exactly 2) - 2(Exactly 3); Total = Russian + Ukrainian + Hebrew + None - (D + E + F) - 2(Exactly 3); 6 = 4 + 3 + 2 + 0 - 3 - 2*G; 6 = 4 + 3 + 2 + 0 - 3 - 2*G; G = 0. P.S. The formula above is explained in details here: ADVANCED OVERLAPPING SETS PROBLEMS . Does this make sense? Does the above answer your query ? Sets.png

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Luca46 · Answer

I like the solution - it’s helpful.

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