Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 1 / c

The outermost braces form a singleton set, regardless of how its one element is nested. Its cardinality is therefore $1$. A set containing the empty set is different from the empty set itself.

A Cartesian product with the empty set is empty: there is no possible second coordinate. Thus the requested cardinality is $0$, independently of the first factor's power set.

The singleton set $\{\varnothing \}$ has cardinality one, so its power set has cardinality two. Taking the Cartesian product with the singleton $\{\varnothing \}$ leaves that cardinality equal to two. The outer power set therefore has $2^2=4$ elements.