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, 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 $i=1,2,...,k$.