I've got a very simple question yet it has been bugging me for ages, WHY/HOW does 0 permutations/arrangements =1 (0!=1)?
If , then , and there exists one map from to ... the empty map !!
Remember how maps are defined: a map from to is a subset of such that, for every , there exists exactly one such that , and we write .
If , then is a subset of (whatever is) that satisfies the assumption since there is no (hence the condition is automatically fulfilled: it is empty). Thus, is a map from to anything. And it is the only one.
It is bijective from to : for every , there is a unique mapped to . Indeed, there is no such , so there's nothing to be checked.
Thus, is the only map from to itself, and it is bijective. As a conclusion, .
This makes sense in many ways, like Prove It illustrated.