Let $A$be a set containing $n$elements. The number of ways of choosing two subsets$P$and $Q$ of  $A$such that 

$P\cap Q=\varphi$ is 

For each $x\in A$, we have four choices;

(i)  $x\in P,x\in Q$

(ii)  $x\in P,x\notin Q$

(iii) $x\notin P,x\in Q$

(iv) $x\notin P,x\notin \mathrm{Q}$

Out of these four choices, last 3 choices imply 

$x\notin P\cap Q$

Thus, $P$ and $Q$ can be chosen in $3^n$ ways so that

$P\cap Q=\varphi$