Question

Let A be a set containing n elements. A subset P of the set A is chosen at random.The set A is reconstructed by replacing the elements of $P$ and another subset $Q$ of A is chosen at random. The probability that $P\cap Q$ contains exactly $m(m<n)$ elements is

• A. $\frac{3^{n.m}}{4^n}$
• B. $\frac{{}^n{C}_m\times 3^m}{4^n}$
• C. $\frac{{}^n{C}_m\times 3^{n.m}}{4^n}$
• D. None of these

Answer 1

Answer: (C)

$\frac{{}^n{C}_m\times 3^{n.m}}{4^n}$

Let A be a set containing $n$ elements.

The number of way in which a subset $A$ cab be chosen

$={}^n{C}_0+{}^n{C}_1+{}^n{C}_2+......+{}^n{C}_n=2^n$

The set $A$ is reconstructed by replacing the elements of $P$ and another subset $Q$ of $A=2^n+2^n=4^n$

Probability that $P\cap Q$ contains exactly $m$ elements $={}^n{C}_m$

$\therefore \frac{{}^n{C}_m\times 3^{n.m}}{4^n}$

Hence, the answer is $\frac{{}^n{C}_m\times 3^{n.m}}{4^n}.$