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.
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