A root system $Φ$ of rank $n$ determines an $n$-dimensional smooth projective toric variety $X(Φ)$ associated with the fan of its Weyl chambers. For the root system of type $A_n$, this variety is the well-known permutohedral variety $X_{A_n}$. Using purely combinatorial methods, we obtain an explicit closed formula expressing the product of Chern classes $c_k c_{n-k}$ as a multiple of the top Chern class $c_n$ in the rational cohomology ring $H^*(X_{A_n};\mathbb{Q})$. The resulting coefficient, which depends only on $k$ and $n$, is given by a closed-form expression. As an application, we compute the Chern number $\langle c_k c_{n-k}, [X_{A_n}] \rangle$.