JEE Advanced · 2010 · 11 Apr 2010 · Paper 2 · Sets Relations and Functions

For each element in S, there are three choices: it can belong to the first subset, the second subset, or neither. This gives $3^4=81$ ordered pairs of disjoint subsets.

Since we want unordered pairs, we must account for the fact that swapping the two subsets gives the same unordered pair, except when one or both subsets are empty.

The number of ordered pairs where both subsets are non-empty is $81-2^4-2^4+1=81-16-16+1=50$ (using inclusion-exclusion: subtract cases where first is empty, subtract cases where second is empty, add back the case where both are empty).

The number of unordered pairs with both non-empty is $50/2=25$.

Adding the cases where at least one is empty: there are $2^4=16$ subsets, so 16 pairs of the form $(A,\varnothing )$, which are the same as $(\varnothing ,A)$ when unordered.

Total: $25+16=41$.

Alternatively, the formula is $\frac{3^n+1}{2}=\frac{81+1}{2}=41$.