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If the question ask about sitting in a line then I can find the way. It is simply $m!n!\binom{m+1}{n}$ or $m!P^{m+1}_{n}$. But how to find when sitting in round table? I need a general formula.

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    Hint: $n$ men can sit around a table in $(n!)/n = (n-1)!$ ways. – 2017-02-08

2 Answers 2

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Arrange the $m$ men in $(m-1)!$ ways.

And arrange the $n$ women (if $n\le m$) in the $m$ positions in $\dbinom{m}{n}$ ways. To take care of order of arrangements of the $n$ women, multiply with $n!$

So, number of ways is

$(m-1)! \times \dbinom{m}{n} \times n!$

So, I think, your given answer is incorrect.

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    I gave the answer for the case when they sit in a row..... Is that wrong also? :| – 2017-02-08
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    Why we aren't dividing by $n$ when counting the permutation of women? I didn't get this :( – 2017-02-08
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    for the row, i think it is correct. – 2017-02-08
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    @12345678, where did i divide with n? – 2017-02-08
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    No.. I asked why you didn't divide by n .... as women were sitting in circular table also – 2017-02-08
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    $n$ women sits in a row $\implies n!$ arrangements. $n$ women sits around a circle $\implies (n-1)!$ arrangements or $n!/n$ arrangements. hope this helps – 2017-02-08
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    That is what you did for the males.... $(m-1)!$... But for the next time.... when women sitting... then why not divide by $n$ ? – 2017-02-08
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    once the males are seated, then we have reference point and each arrangement can be measured. so no need to divide when women sitting – 2017-02-08
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    @Kiran Are you taking into consideration that **no two women are adjacent?** See title of question: *In how many ways m men and n women can sit around a round table, such that no 2 women are adjacent?* – 2017-02-08
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    @amWhy, yes I did. That is why I was trying to place the $n$ women in the $m$ positions where these $m$ positions are separated by males. – 2017-02-08
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    Yes, indeed. +1 The question doesn't confirm it, but let's hope that $n+m$ chairs can fit around the table ;-) – 2017-02-08
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    @12345678 Once the males are selected... Then you don't need to divide by $n$.... Now women will sit in the $m$ empty positions separated by males. But when you try to give $m$ males sit around the table then there are same cyclic permutation like $(1,2,\cdots,m), (2,3,\cdots,m,1)$ so you divide by $m$. but after this for females you dont need this – 2017-02-08
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    @12345678 Better you draw some sample permutation and watch yourself what is going on :) – 2017-02-08
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In a Circular arrangement, Total Permutation is (n-1)!

Here we don't want women to sit together, hence we follow these steps.

Step1: Fix all men first around the table. This can be done in (m-1)!

Step2: Now we have m places in between these men where we can fit available n women. This can be done in $\binom mn$ Ways. And n! ways to arrange women.

Total number of ways = $(m-1)! × \binom mn × n!$ ways.