Given some set $A$, suppose we take the family of subsets of size $n$ as $S_n$, from which we take all tuples of size $m$ as $T_m$.
I'd like to be able to efficiently count the number of tuples in $T_m$ whose union contains the set $A$.
For example, with $A=(1,2,3,4)$, for $n=3$ we have $S_n=((1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4))$ and with $m=5$, among the 1024 5-tuples of $S_n$, 1020 of them when unioned contain the set $A$.
I've manually enumerated/counted a few different combinations but fail to see a consistent pattern.
Ideas?
You can do this using inclusion-exclusion. There are $k:=|A|$ conditions for the union not to contain one of the elements of $A$, and by inclusion-exclusion the number of tuples you want is
$$ \sum_{l=0}^{k-n}(-1)^l\binom kl\binom{k-l}n^m\;, $$
where $\binom kl$ is the number of ways of choosing $l$ elements not contained in the union and $\binom{k-l}n^m$ is the number of $m$-tuples of $n$-subsets formed using the remaining $k-l$ elements.