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

Question

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

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

broall · Answer

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.

GMATinsight · Answer

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

Answer: option D

Mo2men · Answer

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?

Thanks in advance

Mo2men · Answer

I was searching for 15 ways but I could notice my problem. Thanks a lot for your help.

Heseraj · Answer

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

EMPOWERgmatRichC · Answer

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

Mo2men · Answer

Thanks Rich for your kind help and support.

banuchandar · Answer

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.

ScottTargetTestPrep · Answer

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.

Answer: D

bumpbot · Answer

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

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