The complex of injective words of permutations which are not derangements is contractible

Let $D_n\subseteq {\mathrm{\Sigma }}_n$ be the set of derangements in the symmetric group. We prove that the complex of injective words generated by ${\mathrm{\Sigma }}_n\setminus D_n$ is contractible. This gives a conceptual explanation to the well known fact that the complex of injective words generated by ${\mathrm{\Sigma }}_n$ is homotopy equivalent to the wedge sum $S^{n-1}$.