Power set of a set with an empty set

The first one is correct.

This is because $\emptyset$ and $\{\emptyset\}$ are different. The first is an empty set whereas the second is a set whose only element is the empty set.

Both are subsets of the given set. This is because the $\emptyset$ is the subset of every set, and as it happens to be an element of the given set, the set containing it as its element is also its subset.

If a set $A$ is such that $\emptyset\in A$, its power set must necessarily contain these two sets:

• $\emptyset$ (like all other power sets), corresponding to selecting nothing from $A$ (not even $\emptyset$, which is something)
• $\{\emptyset\}$, corresponding to selecting $\emptyset$ only

Therefore only the first of your proposed answers is correct, as you think.

Your suggestions differ by having $\emptyset$ and/or $\{\emptyset\}$ included or not.

• We have $\emptyset\in\mathcal P(X)$ because $\emptyset\subseteq X$ (which would hold for any other $X$ as well)
• We have $\{\emptyset\}\in\mathcal P(X)$ because $\{\emptyset\}\subseteq X$ (which is the case because $\emptyset\in X$ in this specific problem)

Therefore, your first variant is correct (and the other two are incorrect because $\emptyset\ne\{\emptyset\}$).