Question

$A$ is a set consisting of $n$ elements.A subset $P$ of $A$ is chosen.The set $A$ is reconstructed by replacing the elements of $P$. A subset $Q$ of $A$ is again chosen. The number of ways of choosing $P$ and $Q$ so that $P\cap Q=\varphi$ is

• A. $2^{2n}{-}^{2n}{C}_n$
• B. $2^n$
• C. $2^n-1$
• D. None of these

Answer 1

Answer: (D)

None of these

Let P has r elements. It can be chosen in ${}_r^{n}C$ ways. Then Q can be chosen from remaining $n-r$ elements in ${}_0^{n-r}C{+}_1^{n-r}C{+}_2^{n-r}C+..{.}_{n-r}^{n-r}C=2^{n-r}$ ways. Thus, both P and Q can be selected in $({}_r^{n}C\ast 2^{n-r})$ ways.
Thus, the number of ways this can be done = ${\sum }_{r=0}^n({}_r^{n}C\times 2^{n-r})={(2+1)}^n=3^n$ ways.
Hence, (D) is correct.