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 $Q$ 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. $3^n$

Answer 1

The correct option is D

$3^n$

The explanation for the correct option:

Step 1. Let $A=\{a_1,a_2,a_3,\dots .,a_n\}$ For each $a_1\in A(1\le i\le n)$ we have the following four cases

$$\begin{gathered} \left(i\right)a_i\notin Pand{a}_i\in Q \\ \left(ii\right)a_i\in Pand{a}_i\notin Q \\ \left(iii\right)a_i\notin Pand{a}_i\in Q \\ \left(iv\right)a_i\notin Pand{a}_i\notin Q \end{gathered}$$

Step 2. For (i), (ii) and (iii), $a_i\notin (P\cap Q)$

Thus, the total number of ways of choosing $P$ and $Q$ is $4^n$

Out of these four choices, (i) is not favorable for $P\cap Q=Q$

Thus, the number of ways$={\mathbf{3}}^{\mathbf{n}}$

Hence, the correct option is option (D).