Question

A set containing $n$ elements. A subset $P$ of $A$ is chosen. The set $A$ is reconstructed by replacing the element 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. $2^n{-}^{2n}{C}_n$
• B. $2^n$
• C. $2^n-1$
• D. $3^n$

Answer 1

The correct option is D $3^n$

Let $A$= $\{a_1,a_2...,a_n\}$
For each $a_1\in A$ $(1\le i\le n)$
we have the 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 number of ways of choosing $P$ and $Q$ is $4^n$.
Out of these four choices, $\left(i\right)$ is not favourable for $P\cap Q$=$Q$
Thus, the number of ways $=3^n$.