Sum of special type of multinomial coefficients can not be integer.

Question

For the sum $$\sum\frac{(m_1+...m_n-1)!}{m_1!...m_n!}(-1)^{u_i}$$ for arbitrary $u_i$ values, where summation runs over the set of $\{m_1,...,m_n\}$:

1. Each $\{m_1,...,m_n\}$ is unique and,
2. $m_i$-s are odd or $0$,
3. The $\{1,0,...,0\}$ is in the summation set,
4. The shifts are excluded, so if $\{m_1,...,m_n\}$ is in the sum set the $\{m_n,m_1...,m_{n-1}\}$ can not be in a set.

With this conditions I assume that this sum can not be integer.