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

Answer 1

Answer: (C)

For an element $a\in A$ we can have
(i) $a\in p,a\in Q$
(ii) $a\in P,a\notin Q$
(iii) $a\notin P,a\notin Q$
(iv) $a\notin P,a\in Q$
$\therefore$ Exhaustive number of cases $=4^n$
If $m$ elements belongs to $p\cap Q$ then $n-m$ elements can satisfy any of the last three possibilities. So favourable number of cases $={}^n{C}_m{3}^{n-m}\cdot (1{)}^m$