8 chairs r there around a circular table. In how many ways can 2women,6men sit so that women are not togeather

Question

If i do by "total ways - ways in which women togeather" => 7! - 6!2! = 3600

If i do case by case:-
1)women seperated by 1 man: 1man can be chosen in 6ways, women can arrange themselves in 2ways and remaining 5 men can sit in 5! ways. So (6 * 5! * 2)ways.. Something is wrong here..??
2)Women seperated by 2men: ?? i m confused..

Please help me do by "case by case" method..

Answer 1

Ways Women Separate =
Total Ways - Ways Women Together

= 8!/8 - (6!)(2!) = 7! - (6!)(2!) = 5040 - 720*2

= 5040 - 1440 = 3600

You are correct.
_________

Case by Case

Here we need to correct it a bit.

1) Women seperated by 1 man in one direction and 5 men in the other direction: 1 man can be chosen in 6 ways, women can arrange themselves in 2! ways and remaining 5 men can sit in 5! ways.

Number of ways = [6*5!] * 2! = 6! * 2! = 1440

2) Women seperated by 2 men in one direction and 4 men in the other direction: 2 men can be chosen in 6*5 ways, women can arrange themselves in 2! ways and remaining 4 men can sit in 4! ways.

Number of ways = [(6*5)*4!] * 2! = 6! * 2! = 1440

3) Women seperated by 3 men in each direction: 3 men can be chosen in 6*5*4 ways, women can arrange themselves in 2! ways and remaining 3 men can sit in 3! ways.

Number of ways = [(6*5*4)*3! / 2] * 2! = (6!/2) * 2! = 720

The reason we divided by 2 is because there are three men on each side. The two sides are therefore indistinguishable. We don't want to double count them.

There are no more cases.

Case by Case Totals = 1440 + 1440 + 720 = 3600

Which matches the above total.

Answer 2

6 men so
one man in between the 2 women
problem solved
very simple
bye

Answer 3

i thnk 16... im not sure though...