The number of arrangements of n distinct objects in a row is given by n!.
The number of arrangements of n distinct objects in a circle is given by (n-1)!.

The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have: n!/n=(n-1)!

Now, 7 men in a circle can be arranged in (7-1)! ways and if we place 7 women in empty slots between them then no two women will be together. The # of arrangement of these 7 women will be 7! and not 6! because if we shift them by one position we'll get different arrangement because of the neighboring men.

So the answer is indeed 6!*7!.

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Hope it helps.
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7 men can sit around a circular table in (7-1)! ways = 6!
[Logic: no: of asymmetric circular permutations of n objects is (n-1)!.]

Next, all you need to do is seat the women in the vacant 7 slots (b/w the men) which can be done in 7! ways

so, my ans is (6! x 7!) ways

Just one question here.........since you will be seating the women also round the circular table, then why is that the logic of no. of asymmetric circular permutations of n objects does not apply here...?

since we have already arranged the 7 men in a circular fashion, the question, thereafter, ceases to be based on circular permutation.

[Tip :- when you consider such circular scenarios, imagine a "passing-the-parcel" game in either clockwise or anti-clockwise direction. the relative order in which the parcels are passed should be unique among all permutations]

Still not convinced?...have a look at my example in the attachment.

Are you sure about that??

Why are women not considerad also circular???

Yes, the solution given above is correct. Think of it this way:
There are 7 men: Mr. A, Mr. B .....
and 7 women: Ms. A, Ms. B ....
14 seats around a circular table.

You seat the 7 women such that no two of them are together so they occupy 7 non-adjacent places in 6! ways. For the first woman who sits, each seat is identical. Once she sits, each seat becomes unique and when the next woman sits, she sits in a position relative to the first woman (e.g. 1 seat away on left, 3 seats away on right etc)

The 7 men have 7 unique seats to occupy. Each of the 7 seats are unique because they have a fixed relative position (e.g. between Ms. A and Ms. B or between Ms. C and Ms. B etc...). So the men can sit in 7! ways.
Total 6!*7! ways.
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FYI, the answer choices are missing from this question.

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Added the options.
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Thanks Bunuel !

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