Question

If $S=\left\{1,2,3,4\right\}$, then the total number of unordered pairs of disjoint subsets of $S$ is

• A. $25$
• B. $34$
• C. $42$
• D. $41$

Answer 1

The correct option is D

$41$

Explanation for the correct option

Suppose there are two subsets of $S$ such as $A$ and $B$.

As the given set consists of four elements then there are three different ways in which elements in $S$ can be an element of $A$ or $B$ or neither.

To find the number of unordered pairs of disjoint subsets, we use the formula

$\frac{{\mathit{P}}^{\mathit{n}}\mathbf{+}\mathbf{1}}{\mathbf{2}}$, where $P$ is the number of ways and $n$ is the number of elements in $S$.

$\begin{array}{rcl}\therefore \mathrm{Unordered}\mathrm{pairs}& =& \frac{P^n+1}{2}\\ & =& \frac{3^4+1}{2}\\ & =& \frac{82}{2}\\ & =& 41\end{array}$

Hence, the correct option is (D).