IPMAT Indore 2022 · Quantitative Ability · Question 12

The rule. For a finite set X with \left| X \right|=k, the power set has

\left| P\left( X \right) \right|={2}^{k}

Build up one layer at a time, starting from A=⌀ with \left| A \right|=0:

Worth seeing concretely at the start: P\left( ⌀ \right)=\{⌀\} — the empty set has exactly one subset, namely itself — so the first layer has size 1, not 0. That single step is where the question is won or lost.

Answer: 16

Shortcut. The sizes go 0\to 1\to 2\to 4\to 16, each being 2 raised to the previous one. Note the jump at the end: {2}^{4}=16, not {2}^{3}=8 — the exponent is the previous size, which was already 4.

The one trap is the first step. \left| P\left( ⌀ \right) \right|=1, because ⌀ is a subset of itself. Starting from 0 and writing {2}^{0}=1 handles it automatically; starting by “there are no subsets” gives 0 and derails everything after.