Rigorous nature of combinatorics Context: I'm a high school student, who has only ever had an introductory treatment, if that, on combinatorics. As such, the extent to which I have seen combinatoric applications is limited to situations such as "If you need a group of 2 men and 3 women and you have 8 men and 9 women, how many possible ways can you pick the group" (They do get slightly more complicated, but are usually similar).
Question: I apologise in advance for the naive question, but at an elementary level it seems as though combinatorics (and the ensuing probability that can make use of it), seems not overly rigorous. It doesn't seem as though you can "prove" that the number of arrangements you deemed is the correct number. What if you forget a case?
I know that you could argue that you've considered all cases, by asking if there is another case other than the ones you've considered. But, that  doesn't seem to be the way other areas of mathematics is done. If I wish to prove something, I couldn't just say "can you find a situation where the statement is incorrect" as we don't just assume it is correct by nature.
Is combinatorics rigorous?
Thanks

Addressing your main point. Yes combinatorics is a proper discipline of mathematics, and it is done as rigorously as analysis or geometry etc.
Ideas from combinatorics crop up a lot in mathematics and a whole lot in computer science, so it's evolved quite a bit from the basic stuff of permutations, combinations etc.
