= Solution
{c}
We construct the two <transitive models> together, taking care that the later collapse really retains the earlier model. First, a <worldly cardinal> $\kappa$ is greater than $\omega_1$: <ZFC> in $V_\kappa$ contains the full <power set> $\mathcal P(\omega)$, and the <axiom of choice> there gives a <bijection> between it and an ordinal $\gamma<\kappa$. That power set is externally uncountable by <Cantor theorem>, so $\gamma\geq\omega_1$. Infinity rules out the finite cardinals and $\omega$ before this argument is applied.
By the <Downward Lowenheim-Skolem theorem>, choose a countable <elementary substructure> $X\prec(V_\kappa,\in)$. Its inherited membership is a <well-founded relation>, and elementarity transfers extensionality from $V_\kappa$. Apply the <Mostowski collapse theorem> to obtain a <countable transitive model> $M$ of <ZFC>. Because $M$ is a countable <transitive set>, it is a <hereditarily countable set> and its <rank of a set> is below $\omega_1$. In particular, $M\in V_\kappa$.
Now take the <Skolem hull> in $V_\kappa$ of the seed
$$
S=\omega_1\cup M\cup\{\omega_1,M\}.
$$
Call it $H$. The seed has external cardinality $\aleph_1$ and lies inside $V_\kappa$. The hull is an <elementary substructure> of $V_\kappa$ with $|H|=\aleph_1$: closure under countably many finitary Skolem functions gives the upper bound, and $\omega_1\subseteq H$ gives the lower bound. Let $\pi:H\to N$ be its <Mostowski collapse theorem> map. Then $N$ is a <transitive model> of <ZFC>.
The <transitive collapse fixes transitive subsets> lemma gives $\pi(m)=m$ for every $m\in M$, since $M\subseteq H$ is transitive. Since also $M\in H$, the collapse sends that element to $M$ itself. Thus $M\in N$, and transitivity of $N$ yields the required inclusion:
$$
\boxed{M\in N\quad\text{and hence}\quad M\subseteq N.}
$$
This is the construction of <nested transitive models from a worldly cardinal>; the next two parts verify its size and ordinal requirements.
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