= Solution
The forcing
$$
\mathbb P=\operatorname{Fn}(\aleph_1^M\times\mathbb N,2,\aleph_1^M)
$$
is $\aleph_1^M$-closed, so by <closed forcing> it adds no new real numbers. Its generic union can be viewed as a sequence
$$
\langle r_\xi:\xi<\aleph_1^M\rangle,
\qquad r_\xi(n)=\bigcup G(\xi,n),
$$
of old reals. For every $r\in\wp(\mathbb N)^M$, conditions asserting that some unused row equals $r$ are dense: assigning all countably many values of that row is a legitimate condition. Thus the generic sequence surjects $\aleph_1^M$ onto the old set of reals.
In $M$, that set had cardinality $\aleph_2^M$. The forcing therefore collapses $\aleph_2^M$ to $\aleph_1^M$, while adding no reals and preserving $\aleph_1^M$. Consequently
$$
M[G]\models 2^{\aleph_0}=\aleph_1.
$$
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