Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ such that
condition 1: $1 \leq a_i \leq 7$
condition 2: $1 \leq b_i \leq 4$
condition 3: $\sum_{i=1}^{2} {a_i} = 8$
condition 4: $\sum_{i=1}^{2} {b_i} = 5$
Is there a general solution to find the number of integer solutions for an inequality like this given conditions? The way I'm finding the number of solution is by generating all possible solutions and check to see if they satisfy all the conditions or not. This solution is very time-consuming.