>**Proposition 1:** The number of integer solutions of the equation 
$$
\sum_{i=1}^{k}x_i = N
$$
where $x_i\geq n_i$ for $i=1,\ldots,k$, is given by 
$$
\binom{N+k-1-n_1-n_2-...-n_k}{k-1}
$$

Now, to tackle the problem as stated, you need to apply Proposition 1 and invoke the [inclusion-exclusion principle][1], in the following sense:

For $i=1,...,k$, set as 
> $q_i$: the property of one of the solutions of Proposition 1, to satisfy the condition 
$$
x_i> m_i
$$

Then,  if we denote:  
 

 - $N(q_i)$, the number of solutions (provided by Prop. 1) satisfying property $q_i$,  
 - $N(q_i q_j)$, the number of solutions (provided by Prop. 1) satisfying both properties $q_i$, $q_j$, .... etc,  

then we have (again applying Prop. 1) that: 
$$
N(q_1)=\binom{N-1-m_1-n_2-...-n_k}{k-1}, \\ N(q_2 q_3)=\binom{N-1-n_1-m_2-m_3-n_4-...-n_k}{k-1}
$$
etc... 

Now all you need to do, is to apply the inclusion-exclusion principle to determine the number of solutions of Proposition 1, **which have none** of the properties $q_i$ for $i=1,2,...,k$.


  [1]: https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle