The factorial function is traditionally introduced as the number of permutations of n items.
It also crops up as the reciprocal of the n-dimensional volume of the n-dimensional simplex defined by
There’s a routine proof by induction that the volume of an n-simplex is , but it leaves the connection with the number of permutations of n items a little mysterious.
Here’s one way to see why they’re related.
Start by considering the n-dimensional cube of volume one, namely
Now consider the subset
It has volume .
This is because there are subsets of the form where is a permutation of . They all have the same volume (permuting axes doesn’t change volume). And, finally, their disjoint union is the hypercube (modulo subsets of the hyperplanes where , which doesn’t change anything because they have zero volume)
All that’s left is to show that the above subset of the hypercube has the same volume as the n-simplex. One way to do that is to find a volume preserving linear transformation (i.e. with determinant one) that changes one to the other. And there is such a transformation, namely: