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. Continue reading “Volumes and Permutations”