Show that if $P_1 = P_2 = \cdots = P_n = P$ in $\mathbb{R}^n$, then $\text{MV}(P_1,P_2,\dots,P_n) = n! \text{Vol}(P)$.

### Solution:

We are looking for the coefficient of $\prod_{i=1}^n\lambda_i$ in $\mathrm{Vol}(\lambda_1 P + \cdots + \lambda_n P)$, where $P \subset \mathbb{R}^n$. But this is equal to

$$ \mathrm{Vol}((\lambda_1 + \cdots + \lambda_n)P) = (\lambda_1 + \cdots + \lambda_n)^n\mathrm{Vol}(P).$$

When expanded, the number of ways to obtain the monomial $\prod_{i=1}^n\lambda_i$ is the number of ways to distinctly choose one of the variables from each of the $n$ factors, which could be thought of as the number of words on $n$ letters, i.e. the size of the symmetric group, which is $n!$. Thus, $\mathrm{MV}(P,\ldots,P) = n!\mathrm{Vol}(P)$.