We study copies of the classical sequence spaces \(c_0\) and \(\ell _\infty \) in spaces of \(w^*\) - \(w\) continuous symmetric n-linear mappings. More precisely, we consider the Banach space \(\mathcal {L}_{w^*s}(^n X^*, Y)\) of all \(w^*\) - \(w\) continuous symmetric n-linear mappings from \(X^*\) into Y, together with its closed subspace \(\mathcal {K}_{w^*s}(^nX^*,Y)\) consisting of compact operators. We prove that \(\ell _\infty \) embeds in \(\mathcal {K}_{w^*s}(^nX^*,Y)\) if and only if \(\ell _\infty \) embeds into either X or Y. As an application of this, we prove that if \(c_{0}\) embeds in \(K_{w^*s}(^nX^*,Y)\) , then \(\mathcal {K}_{w^*s}(^nX^*,Y)=\mathcal {L}_{w^*s}(^n X^*, Y)\) if and only if only one of the following statements is true: (1) \(c_{0}\) embeds in Y and X has the Schur property,
(2) \(c_{0}\) embeds in X and Y has the Schur property.
