Is there a way to express the condition that exactly $k$ of $N$ boolean variables are active without writing a disjunction of $N$-choose-$k$ terms, i.e., all possible configurations of the $N$ variables where exactly $k$ are active? Is the same true for specifying that at least $K$ of the $N$ boolean variables are active?
I'm aware that this is a problem dealt with in theoretical computer science but I couldn't find any references.