TITLE: Intuition for volume of a simplex being 1/n!
QUESTION [11 upvotes]: Consider the simplex determined by the origin, and $n$ unit basis vectors. The volume of this simplex is $\frac{1}{n!}$, but I am intuitively struggling to see why. I have seen proofs for this and am convinced, but I can't help but think there must be a slicker or more intuitive argument for why this is so than what I have already seen. Any help would be appreciated!

REPLY [8 votes]: Denote the volume of this simplex by $\sigma_n$. Foliating the simplex  with "horizontal" hyperplanes $x_n=z$ $(0\leq z\leq 1)$ and applying Fubini's theorem we obtain
$$\sigma_n=\int_0^1(1-z)^{n-1}\sigma_{n-1}\>dz={1\over n}\sigma_{n-1}\ .$$