If two subsets $\mathrm{A}$ and $\mathrm{B}$ are selected at random from a set $\mathrm{S}$ containing…
If two subsets $\mathrm{A}$ and $\mathrm{B}$ are selected at random from a set $\mathrm{S}$ containing $\mathrm{n}$ elements, then the probability that $\mathrm{A} \cap \mathrm{B}$ $=\phi$ and $\mathrm{A} \cup \mathrm{B}=\mathrm{S}$, is
$\frac{1}{2^n}$
$2^n$
$\frac{1}{2^{\mathrm{n}+1}}$
$\frac{1}{2^n \times 2^n}$
Solution
Given that $S$ contains $n$ elements and two sets $A$ and $B$ are selected.
Two set $A$ and $B$ can be selected in $2^n$ ways.
The no. of ways of selecting two sets such that their union is $S$ and intersection is $2^n$.
Therefore the probability $=\frac{2^n}{2^n \cdot 2^n}=\frac{1}{2^n}$