Given: B={1},find no. of elements of P[P{P(B)}].
If a set has n elements then the powerset of that set will ahve 2^n elements. Each subset is defined by chosing whther or not particular element of the set will go into that subset, so for each element you have two chhoices (independent of the choice you make for any other element). The number of distinct subsets is thus 2^n.
P(B) has 2^1 = 2 elements.
P(P(b)) has 2^2 = 4 elements.
P(P(P(B))) has 2^4 = 16 elements.