You are correct in saying that there are 120 ways to pick the three men and 715 ways to pick the four women. However, the final answer is NOT 835. You do not add those subtotals you got. Instead, you will multiply them.
The actual numeric answer is 120*715 = 85800
Side note: The symbolic answer is C(10,3)*C(13,4)
where the C(n,r) notation refers to the combination formula
C(n,r) = (n!)/( r!*(n-r)!)
in which you referenced earlier. Computing the symbolic answer leads directly to the numeric answer because
C(10,3) = 120
C(13,4) = 715
C(10,3)*C(13,4) = 120*715 = 85800
-----------------------------------------
Here's one way to see why we multiply instead of add:
Consider a tree diagram that consists of 2 branches on the first level, then 3 branches per node on the second level. Overall, there are 2*3 = 6 different paths in which are labeled path A,B,C,D,E,F
This small example is similar to the problem you posted, though the numbers are much larger of course. If you really wanted to carefully lay out the tree diagram, then you'd have 120 different branches for the upper layer. For each segment laid out, you would have 715 additional segments to fill out the second layer. So in a sense, you are adding 715+715+715+...+715 a total of 120 times. A shortcut is to write 120*715, since this is how multiplication is defined (by repeated addition). Not only is this nearly impossible to draw out (unless you have a massive piece of paper and a lot of patience), its very easy to lose track of all the paths possible. Thankfully, such a diagram isn't required to get the numeric answer.
-----------------------------------------
Edit: Your teacher may want you to use a slightly different symbolic notation
n C r instead of C(n,r). I would ask to see which format your teacher prefers.