Why this solution does not work? What is the conceptual gap here?

Since two opening are for women only so two out of  6 are selected in 6c2 ways. Then for the remaining two openings out of 9 men and 4 women , 2 are selected which can be done in 13c2 ways. so total of 1170 ways.

Hi Sourav,
Cases are being repeated in your method. Let me explain it with the help of an example. Say a, b, c, d, e, f are females and 1, 2, 3. . .. . 9 are males.
You are selecting two females so let a, b get selected. Among the remaining people say c and d selected. This is 1 way.
Now let's say while doing 6c2, you get c,d and while doing 13c2 you get a, b. Again you get a, b, c and d. Thus, both cases are same. There will be similar repetitions in your method.
Hope that helps.