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# Last year 26 members of a certain club traveled to England, 26 members

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Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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15 Jun 2016, 01:35
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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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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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16 Jun 2016, 22:12
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Bunuel wrote:
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

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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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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12 Jan 2017, 19:09
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Here's a video explanation for this problem:

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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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16 Jun 2016, 15:10
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
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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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17 Jun 2016, 02:11
3
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Bunuel wrote:
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

Attachment:

Venn Diagram.JPG [ 37.54 KiB | Viewed 26265 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.
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Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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Updated on: 05 Jul 2016, 07:22
5
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

Originally posted by CounterSniper on 05 Jul 2016, 02:57.
Last edited by CounterSniper on 05 Jul 2016, 07:22, edited 1 time in total.
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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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05 Jul 2016, 06:39
VeritasPrepKarishma wrote:
Bunuel wrote:
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

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)

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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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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05 Jul 2016, 23:33
2
6
thangvietnam wrote:
VeritasPrepKarishma wrote:
Bunuel wrote:
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

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)

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.

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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Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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10 Aug 2016, 19:15
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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 25382 times ]

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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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21 Aug 2016, 00:27
CounterSniper wrote:
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

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?
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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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21 Aug 2016, 02:20
1
Neeraj91 wrote:
CounterSniper wrote:
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

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?

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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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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30 Jun 2017, 04:08
1
there can be two formula for this problem. but if you can draw 3 circles , you can remember 3 formula easily.

be careful .
N=A+B+C- AB-BC-AC-2ABC
AB is the number having only two characters
N= A+B+C-AB-BC-AC+ABC
AB is the number having AT LEAST TWO characters.
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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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01 Jul 2017, 20:05
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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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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02 Jul 2017, 01:40
abhisheknandy08 wrote:
Hi Bunuel ,

Could you please help me understand . As where in the question it is given that intersection of all the 3 is 0 .

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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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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03 Jul 2017, 11:50
2
Bunuel wrote:
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

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 20945 times ]

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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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04 Jul 2017, 07:51
3
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Bunuel wrote:
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

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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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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05 Jul 2017, 10:39
VeritasPrepKarishma wrote:
Bunuel wrote:
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

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)

Responding to a pm:

Quote:
Hi, in this problem, you have concluded that members traveling to all three countries is 0, based on "Last year no members of the club traveled to both England and France" from the problem statement. However, in the context of the whole question, it seems as if the implication is "no members traveled to E and F exclusively".

What would the problem statement have to be to rule out the members traveling to all three countries not 0?

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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Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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29 Jul 2018, 02:21
In this question "Last year no members of the club traveled to both England and France," this sentence has a huge implication. It implies that no person traveled to 3 three countries. If anybody traveled to all the three countries, number of members travelling to both England and France would be greater than 0.

f= members that traveled to France only
e=members that traveled to England only
i= members that traveled to Italy only
ef= members that traveled to both England and France = 0
ei= members that traveled to both England and Italy = 6
if= members that traveled to both Italy and France = 11

e + ef + ei = 26
f + ef + if = 26
i + if + ei =32

adding the three equation will give us

e+i+f + 2ef + 2ei + 2if = 84

e+i+f + 2(0) + 2(6) + 2(11) = 84

e+i+f = 50

We found that 50 members traveled to just 1 country. we also know from question that 6 and 11 people traveled to two countries, therefore answer is 50 + 6 + 11 = 67

If this question is still unclear to you, write to me I will try to explain
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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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11 Aug 2018, 18:22
If you are looking for more 3-set Venn Diagram questions to practice on from the Official Guide, I have compiled a list here:

Three-set Venn Diagrams

I think that it is useful to draw out the diagram for 3-set Venn Diagram questions, and this question is a good example of that. The key thing to realize is that there are 0 people in the overlap between England and France, which also means that no one traveled to all 3 countries. Once we see that, we can apply the formula:

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + None

realizing that (all three) = 0 and (sum of 2-group overlaps) = 6 + 11. In addition, we are just looking for the number that traveled to at least one country, so we can set None = 0 and plug in the numbers we were given for each group. Doing that, we get

Total = 26 + 26 + 32 - (6+11) - 2*0 + 0 = 67

Please let me know if you have any questions, and if you would like me to post a video solution!
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Re: Last year 26 members of a certain club traveled to England, 26 members  [#permalink]

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Re: Last year 26 members of a certain club traveled to England, 26 members   [#permalink] 13 Aug 2019, 05:39
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