summation - how to solve $sum _{m=0}^{k-1}mC_{k-1}^{m}C_{N-k}^{m}$?

solving

$$\sum _{m=0}^{k-1} mC_{k-1}^m C_{N-k}^m$$

the solution seems to be

$$\frac {\left( N-2\right) !} {\left( k-2\right) !\left( N-k-1\right) !}$$

according to some clue from the other problem.

struggle with this the whole afternoon, please help.

i tried to extend it, but it's too complicated. i think there should be some smart trick to apply on it to make it easier.

i remember i saw this problem long time ago, but i forgot the solution.

UPDATE: the original suspected solution is wrong. now it's corrected.