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# Anna has to visit at least 2 European cities on her vacation trip. If

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Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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13 May 2017, 14:05
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Difficulty:

65% (hard)

Question Stats:

54% (01:48) correct 46% (01:35) wrong based on 196 sessions

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Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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13 May 2017, 20:43
1
Bunuel wrote:
Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

The question asks the different itineraries, and we need to count the order. For example, visiting city A first and visiting city B later is different from visiting city B first and visiting city A later.

To visit 2 cities from 4 cities, there are $$4 \times 3 = 12$$ different itineraries
To visit 3 cities from 4 cities, there are $$4 \times 3 \times 2= 24$$ different itineraries
To visit 4 cities from 4 cities, there are $$4 \times 3 \times 2 \times 1= 24$$ different itineraries

The total different itineraries are $$12 + 24 + 24 =60$$ different itineraries. The answer is D.
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Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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13 May 2017, 22:32
Bunuel wrote:
Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

Atleast 2 cities to be visited

Case 1: Exactly 2 cities are first chosen and then itineraries are planned in = 4C2*2! ways = 6*2 = 12 ways

Case 2: Exactly 3 cities are first chosen and then itineraries are planned in = 4C3*3! ways = 4*6 = 24 ways

Case 3: Exactly 4 cities are first chosen and then itineraries are planned in = 4C4*4! ways = 1*24 = 24 ways

Total Ways to make itineraries = 12+24+24 = 60 ways

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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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14 May 2017, 04:23
Bunuel wrote:
Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

I did it hard way. I tried to see a pattern for one city. I used London.

Take all combination from London to Paris

L - P
L - P - R
L - P - R -M
L - P - M -R

Take all combination from London to Paris

L - R
L - R - P
L - R - P - M
L - R - M - P

Take all combination from London to Madrid

L - M
L - M - P
L - M - P - R
L - M - R - P

Total ways for one city =12 ways

Because we need to make for FOUR cities =12 * 4 =48 ways.

I would like to know where i went wrong?

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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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14 May 2017, 08:03
nguyendinhtuong wrote:
Mo2men wrote:
Bunuel wrote:
Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

I did it hard way. I tried to see a pattern for one city. I used London.

Take all combination from London to Paris

L - P
L - P - R
L - P - M
L - P - R -M
L - P - M -R

Take all combination from London to Paris

L - R
L - R - P
L - R - M
L - R - P - M
L - R - M - P

Take all combination from London to Madrid

L - M
L - M - P
L - M - R
L - M - P - R
L - M - R - P

Total ways for one city =15 ways

Because we need to make for FOUR cities =12 * 4 =48 ways. 15 * 4 = 60

I would like to know where i went wrong?

I was searching for 15 ways but I could notice my problem.

Thanks a lot for your help.
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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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14 May 2017, 13:53
1
The itinerary should be at least two, meaning the itinerary (x) is equal or bigger than 2. We have three different cases that we need to consider: two cities, three cities and all four cities.

two cities: 4*3= 12
three cities: 4*3*2= 24 and
four cities: 4!=4*3*2=24

sum of three cases: 12+24+24= 60

The correct option: D
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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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14 May 2017, 15:22
2
Mo2men wrote:
Bunuel wrote:
Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

I did it hard way. I tried to see a pattern for one city. I used London.

Take all combination from London to Paris

L - P
L - P - R
L - P - R -M
L - P - M -R

Take all combination from London to Paris

L - R
L - R - P
L - R - P - M
L - R - M - P

Take all combination from London to Madrid

L - M
L - M - P
L - M - P - R
L - M - R - P

Total ways for one city =12 ways

Because we need to make for FOUR cities =12 * 4 =48 ways.

I would like to know where i went wrong?

Hi Mo2men,

Your approach CAN absolutely work here - but you have to make sure that you're considering ALL of the possibilities (and you missed a few).

For example, in your first 'group' (involving London/Paris as the first two locations), you listed the following 4 options:

L - P
L - P - R
L - P - R - M
L - P - M - R

However, you forgot one option...

L - P - M

You made this exact same error in each of your 3 examples, meaning that you missed 3 of the possibilities. If you include those 3 options along with the 12 that you already listed, then you'll have a total of 15 - meaning that there will be (15)(4) = 60 total options (and you'll have the correct answer).

GMAT assassins aren't born, they're made,
Rich
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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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14 May 2017, 15:33
EMPOWERgmatRichC wrote:
Hi Mo2men,

Your approach CAN absolutely work here - but you have to make sure that you're considering ALL of the possibilities (and you missed a few).

For example, in your first 'group' (involving London/Paris as the first two locations), you listed the following 4 options:

L - P
L - P - R
L - P - R - M
L - P - M - R

However, you forgot one option...

L - P - M

You made this exact same error in each of your 3 examples, meaning that you missed 3 of the possibilities. If you include those 3 options along with the 12 that you already listed, then you'll have a total of 15 - meaning that there will be (15)(4) = 60 total options (and you'll have the correct answer).

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

Thanks Rich for your kind help and support.
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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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15 May 2017, 23:55
Am I reading the question stem correctly ?

I am considering the cases where Anna travels to LPR + LPM not in Rome and Madrid included together, meaning if she travels Rome she can not include Madrid in her itinerary.
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Re: Anna has to visit at least 2 European cities on her vacation trip. If  [#permalink]

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07 Oct 2018, 19:25
Bunuel wrote:
Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

To visit at least 2 of the 4 given European cities is to visit 2, 3, or all 4 cities. Since each itinerary must be made up of a different sequence of cities, order is important and thus we have a permutation problem. Let’s determine the number of ways she can visit 2, 3, or all 4 cities.

The number of ways she can visit exactly 2 cities is 4P2 = 4! / 2! = 4 x 3 = 12.
The number of ways she can visit exactly 3 cities is 4P3 = 4! / 1! = 4 x 3 x 2 = 24.
The number of ways she can visit all 4 cities is 4P4 = 4! / 0! = 4 x 3 x 2 x 1 = 24. (Recall that 0! = 1.)

Thus, the total number of ways she can visit at least 2 of the 4 cities is 12 + 24 + 24 = 60.

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Re: Anna has to visit at least 2 European cities on her vacation trip. If   [#permalink] 07 Oct 2018, 19:25
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