A is a set containing n elements. A subset $P$ of A is chosen. The set A is reconstructed by replacing the elements of $P.$ another subset Q  of A is again chosen. Find the number of ways of choosing $P$ and $Q,$ so that  $P\cap Q$ contains exactly $r$ elements.

Let $A=\left\{a_1,a_2,a_3,\dots ,a_n\right\}$

 (i) The r elements in $P$ and $Q$ such that $P\cap Q$ can be chosen out of n is ${}^{n}C_r$ways a general element of A must satisfy one of the following possibilities [here, general element be $a_i(1\le i\le n)]$

  (i) $a_i\in P$and $a_i\in Q$ 

(ii) $a_i\in P$ and $a_i\notin Q$ 

(iii) $a_i\in P$ and $a_i\notin Q$ 

(iv) $a_i\notin P$and $a_i\notin Q$ 

Let $a_1,a_2,\dots ,a_r\in P\cap Q$

There is only one choice each of them (i.e., (i) choice) and three choices (ii), (iii) and (iv) for each of remaining $(n - r)$ elements. Hence, number of ways of remaining elements $=3^{n-r}$ 

Hence, number of ways in which $P\cap Q$ contains exactly r elements $={}^{n}C_r x 3^{n-r}$