Let $S_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. Given a permutation $σ=σ_1σ_2 \cdots σ_n \in S_n$, we say it has a descent at index $i$ if $σ_i>σ_{i+1}$. Let $\mathcal{D}(σ)$ be the set of all descents of $σ$ and define $\mathcal{D}(S;n)=\{σ\in S_n\, | \,\mathcal{D}(σ)=S\}$. We study the Hamming metric and $\ell_\infty$-metric on the sets $\mathcal{D}(S;n)$ for all possible nonempty $S\subset[n-1]$ to determine the maximum possible value that these metrics can achieve when restricted to these subsets.