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Q.
Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to
Permutations and Combinations
Solution:
Let $A$ and $B$ be two subsets of $S$. If $x \in S$, then $x$ will not belong to $A \cap B$ if $x$ belongs to at most one of $A$, $B$. This can happen in 3 ways.
Thus, there are $3^4=81$ subsets of $S$ for which $A \cap B=\phi$.
Out of these there is just one way for which $A=B=\phi$.
As, we, are interested in unordered pairs of disjoint sets, the number of such subsets is
$\frac{1}{2}\left(3^4-1\right)+1=41$