The number of elements in the power set of {{1, 2}, {2, 1, 1}, {2, 1, 1, 2}} is:

Concept

A set is an unordered collection of distinct objects. Two consequences follow directly from this definition: the order in which elements are written is irrelevant, and repeated elements are counted only once. So {a, a, b} and {b, a} denote the same set. Consequently, when one set contains other sets as its members, two member-sets that are equal collapse into a single element of the outer set.

For any set S with n distinct elements, its power set P(S) (the set of all subsets) has exactly 2n elements.

Application

Simplify each inner set using set equality:

1. {1, 2} is already in simplest form: the elements 1 and 2.

2. {2, 1, 1} has a repeated 1 and a reordering, so it equals {1, 2}.

3. {2, 1, 1, 2} repeats both 1 and 2, so it also equals {1, 2}.

4. All three members are the same set {1, 2}. As elements of the outer set they collapse to one, so the outer set is S = { {1, 2} }, which has n = 1 distinct element.

5. Apply the cardinality rule: |P(S)| = 21 = 2.

Cross-check

List the power set explicitly: the only subsets of the one-element set { {1, 2} } are the empty set and the set itself, i.e. P(S) = { ∅, { {1, 2} } } — exactly 2 subsets, confirming the result.

Why 8 is wrong: 8 = 23 arises only if the three written entries are treated as three distinct elements. That ignores set equality — the three entries are literally the same set, so the outer set never has three elements. A video or note that arrives at 8 has counted the entries as written rather than as sets.

Note on the linked video for this question: if its explanation gives the answer as 8 (by treating the three written entries as three distinct elements), that explanation is incorrect for the reason shown above. Trust this written, step-by-step derivation, which follows directly from the definition of set equality and matches the exam's own answer key.