Question

$A$ is a set containing 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. $3^n$
• B. $3{}^2{}^n$
• C. $2{}^2{}^n$
• D. $2^n$

Answer 1

The correct option is A $3^n$

Let $x_1ϵA$
Then the following case arises.
Case I
$x_1ϵP$ and $x_1ϵQ$ ...(not favorable).
Case II
$x_1\notin P$ and $x_1ϵQ$ ...(favorable).
Case III
$x_1\notin P$ and $x_1\notin Q$ ...(favorable).
Case Iv
$x_1\in P$ and $x_1\notin Q$ ...(favorable).
Hence we have 3 favorable cases.
Therefore the number of choosing P and Q such that $P\cap Q=\varphi$
$=3^n$