Question

A is a set containing $n$ elements. A subset $P$ of $A$ is chosen at random, and the set A is reconstructed by replacing the elements of $P$. Another subset $Q$ of $A$ is now chosen at random. The probability that $P\cup Q$ contains exactly $r$ elements, with $1\le r\le n,$ is

• A. $\frac{\left({}^n{C}_r\right)\left(3^r\right)}{4^n}$
• B. $\frac{\left({}^n{C}_r\right)}{4^n}$
• C. $\frac{\left({}^n{C}_r\right)4^r}{3^n}$
• D. None of these

Answer 1

The correct option is A $\frac{\left({}^n{C}_r\right)\left(3^r\right)}{4^n}$

There are four choices for each element of $A$
It can belong to both $P$ and $Q$; $P$ but not $Q$; $Q$ but not $P$; and neither $P$ nor $Q$.
Therefore, since $A$ has $n$ elements, there are $4^n$ ways of choosing the subsets $P$ and $Q$ of $A$.
Now, of the four choices for each element of $A$, it can belong to $P\cup Q$ only $3$ ways.
So, $r$ elements can belong to $P\cup Q$ in $\left({}^n{C}_r\right)3^r$ ways.
Hence, the required probability is $\frac{\left({}^n{C}_r\right)3^r}{4^n}$