Difficult Quant Section 08 April 2021 Question № 2

Hey Greg, I think that this question’s answer choices are wrong. The answer has to be 144 instead of 192.
If we think that E and D should sit together, total choices without considering A and F apart from each other, 5 ! x 2 = 240. However, when you think about when A and F should sit together (then subtracting it from 240) you have to write 4 ! x 4 = 96 because E and D can replace their places as well as A and F can replace their places too. I mean, 4 ! ED AF B C → 4 ! = 24 But both pairs can replace their places.
ED AF B C → DE AF B C → ED FA B C → DE FA B C (4 options). So answer has to be 240 - 4 ! x 4 = 144. Am I wrong?

@gregmat@Leaderboard please take a look. When subtracting from 240, it is not 4!*2, it is 4!*2*2 because within the two groups both DE and AF can switch seats.

I solved it in the conventional method I was taught in, and the answer mentioned by @niyaz.ahmadzada holds.

Here’s my solution.

DE are considered one unit, say x. So we have → A,B,C,X,F

Now if A and F never sit together, so let’s leave them out of the arrangement for now. and arrange B,C,X.
These 3 can be arranged in 3! ways. (PARTIAL RESULT #1)

Now find the gaps between and around BCX and arrange A and F in those gaps. The gaps would be like |B|C|X|. The vertical lines being the gaps where we can put A or F. So now we have 4 gaps.

To seat 2 people in 4 gaps, we first choose any 2 gaps, and then arrange A and F in all possible orders. Choosing 2 gaps out of 4 will be 4C2. (PARTIAL RESULT #2)

To seat 2 people in the chosen gaps we have 2! number of ways. (PARTIAL RESULT #3)

now within the group X, D and E can sit in 2! ways. (PARTIAL RESULT #4)

Finally. the answer would be 3!*4C2*2!*2! which is 144.

You have got 192 as a right answer choice in “Difficult Quant Section 08 April 2021” It was unclear to me and that is why I posted this question here. You are always welcome Greg.