Let $X$ and $Y$ be two disjoint subsets of $S$.
Now, for each of the 4 elements of $S$ we have 3 choices:
- Go to $X$ and not $Y$.
- Go to $Y$ and not $X$.
- Go to neither $X$ nor $Y$.
Thus we get two disjoint subsets $X$ and $Y$ in $3^4 = 81$ ways. But
- we need to count unordered pairs- $ (X = \{1,2\}, Y = \{3,4\})$ and $ (X = \{3,4\}, Y = \{1,2\})$ should not be counted separate.
We can see that every pair $X,Y$ have two orders except $(X = \emptyset, Y = \emptyset)$. So, to count the no. of unordered pairs we can do $$\frac{81 - 1}{2} + 1 = 41.$$