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 offtopic by Andrés E. Caicedo, Stefan Kohl, Monroe Eskew, Igor Belegradek, S. Carnahan♦ Jun 22 '14 at 3:56
This question appears to be offtopic. The users who voted to close gave this specific reason:
 "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andrés E. Caicedo, Stefan Kohl, Monroe Eskew, Igor Belegradek, S. Carnahan

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$\begingroup$ Questions tend to end with question marks. $\endgroup$ – Michael Greinecker Jun 21 '14 at 1:03
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 $\theta$.