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 A is again chosen. The number of ways of choosing P and Q so that P∩Q  contains exactly two elements is

Let A={a1,a2,a3,...........an}.  For any ai∈A,  we may have following situations.    (i)             ai ∈ P,                       ai ∈ Q   (ii)             ai ∈ P,                       ai ∉ Q (iii)             ai ∉ P,                       ai ∈ Q (iii)             ai ∉ P,                       ai ∉ Q ∴P∩Q  contains exactly two elements. Taking 2 elements belonging to case (i) and  (n−2)  elements will belong to case (ii) or  (iii) or (iv) ∴ Number of ways = nC2×3n−2.