There are 13 South American countries, Ann has won a ticket

Question

There are 13 South American countries, Ann has won a ticket to fly from Denver to one South American country and then directly to another South American Country before returning home to Denver again. Assuming there are four airports in each country, how many flights paths can Ann choose from?

A. 13.12.11.10.4.4.
B. 13.12.11.10.4
C. 13.12.4.4.
D. 13.12.4
E. 13.13.13.13

Doubt-

I was able to solve this question, however after looking at the solution, I got confused. 
The question is solved using Permutation formula.

As per my concepts, I use Combination in case I need to do selection and use Permutation in case I need to do arrangement. I know, this is just the basic definition of pnc, however I am able to solve 90% of the question using this definition.

I wanted to understand that why permutation formula is used in the solution. Is it right to say that -

Use Combination in case we need to draw k things out of n identical things
and
use permutation in case we need to selection of one item depends on the previous items. i.e you need to draw one item at a time.

Please clarify

cyberjadugar · Answer

Hi,

My approach is:
There are 13 ways to chose the first destination,
then, 12 ways to chose next destination, with option of 4 airports (13x4)
then, to return he has 4 options to chose the airports

Thus, 13*12*4*4 ways.

Using permutations:
If the solution had written 13P1 instead of 13C1 it won't matter. As per the definition - making arrangements of the combinations gives permutations.
Since there are no arrangement for 1 item, permutations & combinations are equal.

Let me know how the permutations were used in the solution, so that I can give a better thought.

Regards,

imhimanshu · Answer

Thanks for the explanation.

Official Solution says that 
For the first country, Ann has 13 options.
For the second country, Ann has 12 options, coz one is already taken and can not be repeated.

In each country, there are 4 options, so the result are as follows.

# airports for first destination : 13* 4
# airports for second destination : 12* 4

Note that 13P 2 = 13.12(This is what I am trying to understand)

Therefore, total num of flights is 13.12.4.4.
Hence C

cyberjadugar · Answer

Sorry for the mistake, I usually do that :oops:

Will correct the solution.

cyberjadugar · Answer

13P2 = 13*12, true.

or 13P2=13C2*2! (2!, beacuse chosing of same to destination can be done in two different ways, AB or BA)

BrushMyQuant · Answer

Bunuel : Please Untag Probability

Bunuel · Answer

______________
Done. Thank you.

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There are 13 South American countries, Ann has won a ticket

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There are 13 South American countries, Ann has won a ticket

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