I want to find the number of possible solution sets for this problem (so that I can compute probabilities later).
We want to find two matrices $A_{m \times n} = [a_{i,j}]$ and $B_{m \times n} = [b_{i,j}]$, such that,
$$ a_{i,j} \in \{0,1\}, b_{i,j} \in \{0,1\} $$ $$ \forall i, \sum_j{a_{i,j}} = 1 $$ $$ \forall i, \sum_j{b_{i,j}} = K_1 $$ $$ \forall j, \sum_i{(a_{i,j} + b_{i,j})} \leq K_2 $$ $$ \forall i, \forall j, a_{i,j} + b_{i,j} \leq 1 $$
Is there any way I can simplify this, so that it becomes easier to count the solution space?