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if you apply 4C2, you need to understand why. 4C2 means how many ways you have to choose 2 chairs out of 4 chairs but it is not enough.

4C2 is right if you think this way:

1) let's choose any 2 seats out of 4. 2) For chosen 2 seats we pick left seat for A and right seat for B.

but that red colored constraint is a violation on the very definition of combination ....definition of combination talks about group ...it does not bother about specific order....is not it ? ....so, we really cant write down 4C2 here as per the combination rule .

I am happy with your earlier solution.

I found in a book ...they used 4C2 in the solution . I object this ...because this is against the definition of combination ( as there is a constraint).

But I observe you did not use this 4C2 thingies and hence I liked your solution.

There are four places and two of them want to sit in only one way. If we combine two (A & B) places into one, then there are 6 ways to occupy three places and 1 way to arrange the A and B.