Given an inner model $W subset V$ and a regular cardinal $kappa$, we consider two alternatives for adding a subset to $kappa$ by forcing: the Cohen poset $Add(kappa,1)$, and the Cohen poset of the inner model $Add(kappa,1)^W$. The forcing from $W$ will be at least as strong as the forcing from $V$ (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from $V$ to fail to be as strong as that from $W$. The results are generalized to $Add(kappa,lambda)$, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.
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