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Last year 26 members of a certain club traveled to England, 26 members traveled to France, and 32 members traveled to Italy. Last year no members of the club traveled to both England and France, 6 members traveled to both England and Italy, and 11 members traveled to both France and Italy. How many members of the club traveled to at least one of these three countries last year?
A) 52
B) 67
C) 71
D) 73
E) 79
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A) 52
B) 67
C) 71
D) 73
E) 79
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Bunuel
n(E or F or I) = n(E) + n(F) + n(I) - n(E and F) - n(F and I) - n(I and E) +n(E and F and I)
n(E or F or I) = 26 + 26 + 32 - 0 - 11 - 6 + 0
Why is the number of people who traveled to E and F and I 0? Because no one traveled to both England and France. So obviously, no one would have traveled to England, France and Italy.
n(E or F or I) = 67
Answer (B)
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GMAT 1: 780 Q51 V46
Posts: 58
Here's a video explanation for this problem:
thangvietnam
These are two different formulas:
Total = 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(No Set)
Total = n(A) + n(B) + n(C) – n(Only A and B) – n(Only B and C) – n( Only C and A) - 2* n(A and B and C) + n(No Set)
n(A and B) in first formula includes the elements falling in n(A and B and C)
n(Only A and B) in second formula does not include the elements falling in n(A and B and C). It has element which fall in A and B only, not in C.
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General Discussion
Senthil1981
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Answer is B) 67
Draw Venn diagram to represent this and you'll notice that people traveled to more than one country but not all three. The question is same as how many people traveled ?
So 26 + 26 + 32 - (11 + 6) = 67
Draw Venn diagram to represent this and you'll notice that people traveled to more than one country but not all three. The question is same as how many people traveled ?
So 26 + 26 + 32 - (11 + 6) = 67
Bunuel
Attachment:
Venn Diagram.JPG [ 37.54 KiB | Viewed 111353 times ]
a+b= 6
b+c = 11
Refer England team: a+b+ (persons who only went to England) = 26
Persons who only went to England = 26 - (a+b) = 26-6 = 20
Refer French team: b+c+(persons who only went to France) = 26
Persons who only went to France = 26 - (b+c) = 15
Refer Italian team: a+b+c+f = 32
Now with above info.
Members of the club traveled to at least one of these three countries = Nothing but summation of all values (take only once)
Leftmost circle (England) : (20+a+b)
French: 15+c (all remaining already covered in previous circle)
Italy: f (all other variables taken in previous two circles)
(20+a+b)+(15+c)+f will give the answer
20+15+ a+b+c+f = 35+32 = 67
B is the answer.
Posts: 609
Updated on: 05 Jul 2016, 06:22
Originally posted by CounterSniper on 05 Jul 2016, 01:57.
Last edited by CounterSniper on 05 Jul 2016, 06:22, edited 1 time in total.
Last edited by CounterSniper on 05 Jul 2016, 06:22, edited 1 time in total.
to england=26
to france = 26
to italy=32
england and france =0
england and italy = 6
france and italy =11
all 3=0
neither=0
total=26+26+32-(0+6+11)+0 +0
= 67
to france = 26
to italy=32
england and france =0
england and italy = 6
france and italy =11
all 3=0
neither=0
total=26+26+32-(0+6+11)+0 +0
= 67
Posts: 513
VeritasPrepKarishma
i am confused, pls, help me.
I think the formulae is
N= E+F+I - N(E AND F) - N (F and I)- N (I and E) - 2 N (E AND F AND I)
we need minuse two times the number of member who travel to 3 countries- in this case this number is zero.
but in general formulae, we need minus, not plus .
pls, explain, i am confused.
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Expert reply
Attached is a visual that should help. Note that 3! is 6, which confirms that there are only 3 overlapping sets of 2 to account for.
Attachments
Screen Shot 2016-08-10 at 7.14.11 PM.png [ 87.55 KiB | Viewed 105564 times ]
CounterSniper
I did it the same way as well, however I wasn't sure of one thing. How do we know that all three = 0? If nothing is mentioned about it, is it assumed to be 0?
Neeraj91
No, it is not assumed. If you take ALL three as Non Zero, then we will have few who travel to both England and France. But the question stem says, we don't have anyone who traveled to both England and France.
I hope its clear now.
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Hi Bunuel ,
Could you please help me understand . As where in the question it is given that intersection of all the 3 is 0 .
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Could you please help me understand . As where in the question it is given that intersection of all the 3 is 0 .
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This is explained in Karishma's post above: "Why is the number of people who traveled to E and F and I 0? Because no one traveled to both England and France. So obviously, no one would have traveled to England, France and Italy".
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Bunuel
We are asked for members who traveled to at least one of these three countries, it can also include members who have traveled two.
As per the attachment, the calculation will be
\(20 + 6 + 15 + 11 + 15 = 67\)
Hence, Answer is B
Attachments
GMAT.PNG [ 8.03 KiB | Viewed 100016 times ]
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Bunuel
We are given the following:
England travelers = 26
France travelers = 26
Italy travelers = 32
England and France travelers = 0
England and Italy travelers = 6
France and Italy travelers = 11
Although it’s not stated directly, we can determine that 0 people traveled to all 3 countries because 0 people traveled to both England and France.
In determining how many people traveled to at least one country, we are actually determining the total number of travelers, since each traveler did travel to at least one country. We can do this with the following formula:
Total travelers = England + France + Italy - sum of (exactly two countries) - 2 times (all three countries)
Total travelers = 26 + 26 + 32 - (6 + 11 + 0) - 2(0)
Total travelers = 84 - 17 - 0 = 67
Thus, 67 people traveled to at least one country.
Answer: B
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VeritasPrepKarishma
Responding to a pm:
Quote:
This is what the problem states: "Last year no members of the club traveled to both England and France"
Both (England and France) is 0.
So no one could have travelled to both countries. Hence, travelling to all 3 is not possible.
Had the question said: "Last year no members of the club traveled to England and France only"
then all three was possible.
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VeritasKarishma
Hi karishma VeritasKarishma
I am struggling with your implication regarding this statement : "Last year no members of the club traveled to both England and France", implies travelling to all 3 is not possible
Specifically, If i were to pose the following scenario
-- There are the three groups of people (26 English / 26 French/ 32 Italian)
-- Last year no Englishmen, traveled to France
Question : in this scenario, you would obviously mark "Nil" in the image and not mark "Nil" automatically in space B in the image [Reasoning : maybe there are Italians who have traveled to England and France]
Why the difference in logic in this scenario ?
Attachments
Overlapping sets.png [ 158.76 KiB | Viewed 75768 times ]
The question does not say that Englishmen travel to another country. The people we are talking about are, say Australian.
"Last year 26 members of a certain club traveled to England, 26 members traveled to France, and 32 members traveled to Italy."
If some of them already belong to one of the three countries mentioned, the question makes no sense. So say we have a club of Australian people and these are their travel histories. If no one travelled to both England and France, then obviously no one travelled to England, France and Italy.
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Bunuel
Answer: Option B
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IMO B.
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