Answer
a) 8
b) 16
c) 2
Work Step by Step
Recall the power set of an n element set has $2^n$ elements.
a) $\{ a, b, \{a,b\}\}$ has three elements, so $\mathcal{P}(\{a,b,\{a,b\}\})=2^3 = 8$.
b) $\{ \emptyset , a, \{a\}, \{\{a\}\}\}$ has four elements, so $\mathcal{P}(\{ \emptyset , a, \{a\}, \{\{a\}\}\}) = 2^4 = 16$.
c) We know $\mathcal{P}(\emptyset ) = \{\emptyset \}$, which is a one element set. So $\mathcal{P}(\mathcal{P}(\emptyset )) = \mathcal{P}(\{ \emptyset \}) = 2^1 = 2$.