(1) The most fundamental is the principle that says "counting makes sense": in set theory language, given two finite sets A and B (this is true also for infinite sets), if there is a bijective function between them, them they have the same number of elements. This principle is fundamental in Combinatorics because, by constructing a bijection between a set of things that we know how to count (that is, we know how many of them there are) and a set of things that we don't know how to count then, by using this principle, we know that they have the same number of elements. (Proofs of this type are called "bijective proofs" in Combinatorics, and they are considered most insightful proofs).
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