Complexity/Hardness of a generalization of an Inclusion/Exclusion problem I would appreciate some help in determining the complexity/hardness of an inclusion/exclusion problem described in Wikipedia: https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle#A_generalization

Namely, given a family of subsets $A_1$, $A_2$, ..., $A_n$ of a universal set $S$, I need to calculate the number of elements of $S$ which appear in exactly some fixed $m$ of these sets.

Would that be #P?

Thank you.
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.