IIT-JEE 2010 · Paper 2, Mathematics question 21

Ordered pairs $(X,Y)$ of disjoint subsets: each element goes to $X$, to $Y$, or to neither: $3^4=81$.

Unordered. Every pair with $X\ne Y$ is counted twice; the only pair with $X=Y$ (disjoint) is $(\varnothing,\varnothing)$. So the number is $\dfrac{81-1}2+1=41$.

(D)