Sum of products of binomial coefficient

Question

The multinomial theorem states that \begin{align} (x_{1} + \dots + x_{m})^{n} = \sum_{k_1 + \dots + k_{m} = n} \binom{n}{k_1, k_2, \dots, k_{m}} x_{1}^{k_{1}} \cdots x_{m}^{k_{m}} = \sum_{k_1 + \dots + k_{m} = n} \frac{n!}{k_1!k_2!\cdots k_m!} x_1^{k_1} \cdots x_{m}^{k_{m}} \end{align} I need to evaluate this sum here \begin{align} \sum_{j=0}^{n-1} \binom{n-1}{j} \sum_{k_1+\cdots+k_m=j} j!\binom{n}{k_1}\cdots\binom{n}{k_m}\frac{1}{(-1)^{k_1}\cdots(-m)^{k_m}} \end{align} which differs from the multinomial theorem with $\frac{1}{(n-k_1)!\cdots(n-k_m)!}$, so I actually can not use the multinomial theorem. The sum above by right should have evaluated to $1$, yet I have no idea where to start, any help would be appreciated, thanks in advance.