TITLE: Interpreting $n!$ as the volume of a $1 \times 2 \cdots \times n$ box
QUESTION [16 upvotes]: Q. Are there relationships or proofs that are illuminated
  by viewing $n!$ as the volume of a $1 \times 2 \cdots \times n$ box
  in $n$-dimensions?

I cannot think of any, but perhaps they exist...?

REPLY [2 votes]: Consider the matrix $${\bf A_n} = \left(\begin{array}{cccc}1&0&\cdots&0\\0&2&\cdots&0\\\vdots&\vdots&\ddots&0\\0&0&0&n\end{array}\right)$$ 
It's determinant is $n!$. Furthermore 
$$\det({\bf A_m}+k{\bf I_m}) = \frac{(m+k)!}{k!}$$
Which illuminates some relations to combinatorics (permutations).