well, you can do this problem with a lot less work than that.
no information about the number of men ("y"), so there are an infinite number of possibilities.
x and y could be, say, 5 and 4. or they could be 1,000,000 and 999,999. or any of infinitely many other possibilities. these are clearly going to give different final answers, so, insufficient.
from statement 1 you can tell that you're going to get A SPECIFIC # of women.
i.e., whenever you add more women, there are more ways to choose 3 of them. so, if you have a specific # of groups possible (here 56), that fixes the number you're choosing from. since this is data sufficiency, you don't have to find that actual number.
then, from statement 2, you'll also get a specific # of men.
thus you'll have specific numbers all around, so you'll be able to get a specific value for the solution.
as far as your slot method:
women can be chosen in (x+2) * (x+1) * (x) ways
and men can be chosen in (y) * (y-1) ways
this setup is incorrect.
a "panel" is a situation in which order doesn't matter (since there is no particular order, and there are no distinct positions, on a "panel"). so, your product for the women should be divided by 3!, and your product for the men should be divided by 2!.