Complexity/Hardness of a generalization of an Inclusion/Exclusion problem
For each element in $S$, count the number of times it is a member of $A_i$, $1\le i\le n$. If that number is $m$, we add 1 to the tally. The final tally is what we wanted. 

The above algorithm takes $O(tn)$ time if the computational cost to check membership is $O(1)$. It takes $O(ttn)$ time if the computational cost to check membership proportional to the size of the set, which is at most $t$. 

So you can see the problem is in $P$. It has nothing to do with that general inclusion–exclusion principle.