forcing and set theory [closed]

Let $\kappa , \lambda , \theta$ be infinte cardinals de $M$ represents a c.t.m, and $P=Fn( \kappa\ \times \omega,2)$. Show that $(\lambda^{\theta})^{M[G]}\leq((max( \kappa, \lambda)^{\theta})^{M}$. as building function names of $\theta$ in $\lambda$

closed as off-topic by Andrés E. Caicedo, Stefan Kohl, Monroe Eskew, Igor Belegradek, S. Carnahan♦Jun 22 '14 at 3:56

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Any function $f:\theta\to\lambda$ in the extension can be completely described by giving, in the ground model, for each $i\in\theta$, a "nice name" for $f(i)$. A nice name amounts to a maximal antichain of conditions that decide particular values (in $\lambda$) for $f(i)$. So what you need to do is count (1) maximal antichains in your forcing (remember that it has the countable antichain condition), (2) count how many ways there are to label a maximal antichain with members of $\lambda$, and (3) count how many ways there are to assign such a labeled antichain (= nice name) to each element of $\theta$.