Question 1070134
The way to do this is to first recognize that 2 is the least number of black buttons, because one will pick 9, and the other 7 must be brown.  If one picks 1, then it is not possible to have 8 brown buttons, only 7.
So the first term in the numerator is 7C2*7C7, or 21.
7C3*7C6 is the second possibility and 35*7=245
7C4*7C5 is the third possibility and 35*21=735
7C5*7C4 is the last possibility and is also 735
Those 4 add to 1736
1736/2002=124/143
Notice how the combinatorials add to 14C9; the first (top) part adds to 14, the last (bottom) part to 9.