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…

Q: 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 chosen P and Q so that P ∩ Q = φ is

(A) 22n2nCn

(B) 2n

(C) 2n – 1

(D) 3n

Sol. Let A = { a1, a2 , a3 , . . . , an} . For ai ∈ A, we have the following choices:

(i) ai  ∈ P and ai ∉ Q

(ii) ai ∈ P and ai ∉ Q

(iii) ai ∉ P and ai ∈ Q

(iv) ai ∉ P and ai ∉ Q

Out of these only (ii) , (iii) and (iv) imply ai  ∉ P ∩ Q. Therefore, the number of ways in which none of a1, a2, . . .an belong to P ∩ Q is 3n.

Hence (D) is the correct answer.