Question

Four men and four women are seated around a round table. If there are two women do not want to sit next to a particular man, how many arrangments are possible?

Answer 1

I just dont understand arrangments in a circle :/

Answer 2

In circular Arrangement We Fix One...Permenent..))

Answer 3

is one solution, assuming that W1 does not want to sit with M2.
Now you can try to find all solutions.

Answer 4

The "MAN" is the man that nobody wants to sit by, so he make him the one that stays still. Then the 2 seats around him will only have 5 people that could sit by him
(8 - himself - 2women). then each other seat will have starting from 5 all the way to 1 people that can sit in each seat. Then just multiply all of them=1x5x5x5x4x3x2x1 (the first "1" is the "MAN" because he will be the only can be the only one to sit in that seat.

Answer 5

My initial diagram was for 2 men and 2 women, and does not apply to the present problem.

@wizguy @omniscience
I suggest the following change

since once one of the five are seated, there are only four candidates left.
This gives a total of 5*4*(5!) = 2400 ways.

We can also do this combinatorically as a check.
Let the first man sit, there are then 7! ways to sit the rest, including the undesirable ways.
Now try to remove the undesirable ways.
The undesirable ways fall into two cases:
He has both women W1 and W2 sitting on his sides: 2!(5!) ways.
He has only one of W1 or W2 sitting on either side: 4(6!-5!) ways

So total acceptable ways:
7!-4(6!-5!)-2!(5!)=2400