Hello Sharmishtha,
In the Manhattan GMAT math workbooks, they have the idea of creating anagrams with the problem to help with counting the number of ways.
- If you want a total count, you just take the factorial.
- If you have repeats, or things "stuck together" take the factorial, divided by each "stuck together" factorial.
In our problem, we will find the following:
-Total number of ways they can be placed. =
P(Total)
-Total number of ways they can be placed WHERE Couple-1 is together. =
P(C1)
-Total number of ways they can be placed WHERE Couple-2 is together. =
P(C2)
-Total number of ways they can be placed WHERE BOTH Couple-1 and Couple-2 are together. =
P(Both) =
P(C1 ∩ C2)
P(Total) =
A B C D E
= 5*4*3*2*1
= 5!
= 120
-There are five people, after placing the first, there are four people, etc.. Multiplication Rule applies, and we get 5! possible ways to arrange them.
P(C1) =
A A B C D
= (5!) / (2!)
= 120 / 2
= 60
Couple 1 sits together, so they are "stuck together" thus we divide by the factorial of the number of A's in our anagram (2 A's, so 2!). There are 60 ways to arrange the 5 people where couple 1 is sitting next to each other.
P(C2) =
A B C C D
= (5!) / (2!)
= 120 / 2
= 60
Same as above. Couple 2 sits together, so there are 2 C's, so 60 possible ways to arrange them with that seating.
P(C1 ∩ C2) =
A A B B C
= (5!) / (2!)(2!)
= 120 / 4
= 30
Both Couple 1 and 2 are sitting next to each other, hence we divide by 2! twice. 30 ways to arrange the 5 people and still have both couples sitting next to their respective partners.
The formula for Union is P(C1 U C2) = P(C1) + P(C2) - P(C1 ∩ C2)
P(C1 U C2) = 60 + 60 - 30
= 120 - 30
= 90
So there are 90 ways to arrange these 5 people where: C1 is together, C2 is together, or both C1 and C2 are together.
SO, out of the 120 ways, 90 ways they are together. So they ways they are not together is:
120 - 90 = 30.
The probability of neither of them sitting together is
= (# of ways they are not together) / (total # of arrangements)
= 30 / 120
= 3/12
= 1/4
The answer is B.
Hope this helps! If you have any doubts, please let us know!
--Rishi
