The first name and the last name of 5 people are written in two tables

Question

https://gmatclub.com/forum/download/file.php?id=54941 The first name and the last name of 5 people are written in two tables above, in a jumbled order. For example, the last name of John may be Garth. Lisa, who doesn’t know the correct first name-last name pair for any of the 5 people in the table, is asked to create 5 first name – last name pairs using each first name and last name in the tables above only once. What is the probability that the pairs she creates includes the correct first name-last name pairs of 2 people in the table? A. 1/120 B. 1/360 C. 1/240 D. 1/60 E. 1/6 1.png

Archit3110 · Answer

for 5 people getting exactly correct their surnames ; 1/5! ; 1/120
and for 2 people ; 5*4 ; 20 ways
so P ; 20/120 ; 1/6
OPTION E

NitishJain · Answer

VeritasKarishma : Help, please.

bonbonzzz · Answer

doesn't make sense. someone help please?

KarishmaB · Answer

The total number of ways in which we can make first name-lastname pairs is 5! = 120 (There are 5 spots and 5 last names)

Out of these 120, there is one arrangement in which all pairs are correct. There are some arrangements in which no pair is correct, in some only 1 pair is correct, some in which 2 pairs are correct and some in which 3 are correct.

We need those arrangements in which 2 pairs are correct. Pick 2 last names in 5C2 ways and put them in their correct place. 
Now we have 3 last names to be put in 3 spots. Say Jordan Johnson and Sam Jones are correctly paired. 
We are left with Smith, McGill and Garth. 
To put next to John, we have 2 options (excluding his actual last name). Say we put Garth. 
Now we have 2 last names which can be arranged in only 1 way between the two firstnames so that both are incorrectly placed. 
Number of ways of having exactly 2 correct pairs = 5C2*2 = 20 ways

Probability = 20/120 = 1/6

Answer (E)

bonbonzzz · Answer

Hi Karishma,

thanks so much for the explanation, but I'm sorry I got lost right at the beginning.

Do you mean that the total number of ways to arrange 5 names Lisa has created is 5!? To me, there are only 25 ways you can make names out of the two columns - 5x5 possible name combinations.

Secondly, why are we concerned with number of ways the 5 combo Lisa came up with can be arranged, instead of the possible names available? Shouldn't the possibility of her getting one right name be 5/25 or 1/5? the second one would be 4/24. Of course then we'd have to figure out the variations of placement of the correct names.

Thanks!

mSKR · Answer

Total number of pairs possibe
5*4*3*2*1= 120

Only 2 correct:
1*1*2*1*1
correct*correct * wrong* wrong*wrong

now these 2 correct can be of any 5 people , so 5 c2= 10 ways

so total probability of choosing any 2 correct pairs = 10*2/120 = 1/6

How about this approach@VeritasKarishma@@VeritasKarishma

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The first name and the last name of 5 people are written in two tables

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