Consider the birthday problem. Given $N$ people, how many ways are there for there to exist some pair of people with the same birthday?
Enumerating the possibilities quickly becomes tedious
However, the complement problem (Given $N$ people, how many ways are there for no one to have the same birthday?) is trivial.
In fields like probability, this has obvious applications, due to the "complement law":
if $A \cup A^c = S$, where $S$ is the entire sample space, then $$P(A) + P(A^c) = 1 \implies P(A) = 1 - P(A^c)$$
In general, this pattern is very common. Intuitively, I sense:
somehow, the complement problem is asking for a lot less information
if one has something like the "complement law" in probability, then in some restricted scope of problems, the "complement law" gives in some sense, the "same amount of information"
What do mathematicians call what I am getting at here? Am I overblowing how common a trend it is?