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 $={ }^{ n } C _{ m } 3^{ n - m } \cdot(1)^{ m }$