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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)   22n 2nCn 

(B)  2n

(C)  2n - 1

(D)  3n

1 Answer

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Correct option (D) 3n

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 .

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