Q75 / 160 · AP EAMCET 2023 (15 May Shift 1) · Mathematics

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

  • Correct answer
    A
    $\frac{1}{2^n}$
  • B
    $2^n$
  • C
    $\frac{1}{2^{\mathrm{n}+1}}$
  • D
    $\frac{1}{2^n \times 2^n}$

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

Hence, the correct answer is option: 'A'.

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