I'm trying to find a solution for exercise G1 of Chapter VII in Kunen's "Set Theory - An Introduction to Independence Proofs".
The question is in the context of Cohen forcing. We start with a c.t.m. $M$ for a sufficiently large segment of ZFC. In the ground model we fix three infinite cardinals $\kappa,\lambda,\vartheta$. The notion of forcing is $P := \mathrm{Fn}(\kappa\times\omega, 2)$. One now has to prove that $$(\lambda^\vartheta)^{M[G]} = ((\max\{\lambda, \kappa\})^\vartheta)^M$$ for any generic filter $G\subseteq P$.
I'm guessing that the argument uses the fact that $P$ has c.c.c. and since $|P| = \kappa$ there are at most $\kappa^{\aleph_0}$ many antichains in $P$. However, I do not see how to start the argument.
Would someone be so kind as to give me a hint?