How can I compute the number of solutions to an under-constrained equation over some integers?

Question

I have system of under-constrained equations over a fixed set of integers, defined as follows, how can I find the number of solutions to this problem? A computational reference will do (in-fact preferred).

$\alpha_{ij} \in \{0,1,2,3,4\}$. $i$ goes from $1$ to $5$ and $j$ goes from $1$ to $4$

the constraint is given as follows: $$\frac{\sum_{i,j}\alpha_{ij}}{2} = 10$$ $$\sum_j\alpha_{ij} = 4$$