= Solution
{c}
A <cardinal-preserving forcing> over $M$ leaves every ground-model <cardinal number> a cardinal in every <generic extension>. In the present notation, for every $M$-generic filter $G$ and every ordinal $\kappa\in M$,
$$
\boxed{M\models\text{“}\kappa\text{ is a cardinal”}\quad\Longrightarrow\quad M[G]\models\text{“}\kappa\text{ is a cardinal”}.}
$$
Set <forcing> preserves ordinals, and an old bijection witnessing that an ordinal is not a cardinal remains available. Thus this condition also means that the two models have exactly the same cardinals. Cardinality here is computed inside the respective models, not from their external sizes.
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