Question

A is 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 way of choosing P and Q so that P Q =$\varphi$ is :-

• A. $2^{2n}{-}^{2n}{C}_n$
• B. $2^n$
• C. $2^n-1$
• D. $3^n$

Answer 1

Answer: (D)

$3^n$

Let A = $a_1,a_2,a_3,....a_n$. For $a_1$ $ϵ$ For $a_1ϵ$A we have following choices:
(i) $a_1ϵ$ P and $a_1ϵ$ Q
(ii) $a_1ϵ$P $\notin$
(iii) ${}_1\notin Pand{a}_1ϵQ$
(Iv)${}_1\notin Pand{a}_1\notin Q$
Out of these only (Ii), and (iii) and (iv) imply $a_1\notin P\cap Q$ therefore, the number of ways in which none of $a_1,a_2.....a_n$ belong $P\cap Q$ is $3^n$.