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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 chosing P an

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 chosing P and Q, so that P \cap Q contains exactly two elements is

Option: 1

9{ }^n C_2


Option: 2

3^n-{ }^n C_2


Option: 3

2^n C_n

 


Option: 4

none of these


Answers (1)

best_answer

A=\left\{a_1, a_2, a_3, \ldots, a_n\right\}

\\(i) a_i \in P, a_i \in Q\\ (ii) a_i \notin P, a_i \notin Q\\ (iii) a_i \notin P, a_i \in Q\\ (iv) a_i \notin P, a_i \notin Q\\

P \cap Qcontains exactly two elements, taking 2 elements in (i) and (n-2) elements in (ii) or (iii) or (iv)

\therefore \text { Number of ways }={ }^n C_2 \times 3^{n-2}

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