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) 2^{2n} – ^{2n}C_{n}

(B) 2n

(C) 2n – 1

(D) 3n

Sol. Let A = { a_{1}, a_{2} , a_{3} , . . . , a_{n}} . For a_{i} ∈ A, we have the following choices:

(i) a_{i} ∈ P and a_{i} ∉ Q

(ii) a_{i} ∈ P and a_{i} ∉ Q

(iii) a_{i} ∉ P and a_{i} ∈ Q

(iv) a_{i} ∉ P and a_{i} ∉ Q

Out of these only (ii) , (iii) and (iv) imply a_{i} ∉ 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.