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

What you phrased as "can you find a situation where the statement is incorrect" is better known as proof-by-contradiction in the realm of propositional logic. Proof by contradiction is a fairly common approach to proving statements about combinatorics and other areas of study in discrete mathematics--typically after a few initial lemmas and/or theorems have been introduced. This is in contrast to a tautological proof. Solving discrete math problems certainly requires a paradigm shift in thought when compared to algebra, because ultimately not everything is always completely quantifiable. 
