Question

Let $S$ be a set containing $n$ elements and we select two subsets $A$ and $B$ of $S$ at random, then the probability that $A\cup B=S$ and $A\cap B=\varphi$, is.

• A. $2^n$
• B. $n^2$
• C. $\frac{1}{n}$
• D. $\frac{1}{2^n}$

Answer 1

Answer: (D)

$\frac{1}{2^n}$

Given that $S$ contains $n$ elements and two sets $A$ and $B$ are selected.

Two sets $A$ and $B$ can be selected in $2^n$ ways.

The number of ways of selecting two sets such that their union is $S$ and intersection is nullset is $1$.

Therefore the probability is $\frac{1}{2^n}$.

So the correct option is $D$.