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. Find the probability that $P\cup Q$ contains exactly $r$ elements, with $1\le r\le n$.

• A. $P\left(E\right)=\frac{{{}^{n}C}_r{3}^{r-1}}{4^n}$
• B. $P\left(E\right)=\frac{{{}^{n}C}_r{2}^r}{4^n}$
• C. $P\left(E\right)=\frac{{{}^{n}C}_r{3}^r}{4^n}$
• D. $P\left(E\right)=\frac{{{}^{n}C}_r{(-1)}^r}{4^n}$

Answer 1

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

Let $A=\{a_1,a_2,a_3,...a_n\}$
For each $a_i\in A(1\le i\le n)$
we have following four cases:
i) $a_i\in P$ and $a_i\in Q$
ii) $a_i\notin P$ and $a_i\in Q$
iii) $a_i\in P$ and $a_i\notin Q$
iv) $a_i\notin P$ and $a_i\notin Q$
Thus the total numbers of ways of choosing $P$ and $Q$ is $4^n$
and choosing exactly $r$ elements in $(P\cup Q)$ is ${}^n{C}_r{3}^r$
Therefore required probability $=\frac{{}^n{C}_r{3}^r}{4^n}$