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Question

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. Let nC2×kn2 be the number of ways of choosing P and Q so that PQ contains exactly two elements.Find k ?

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Solution

Let A={a1,a2,a3,...,an}
The two elements P and Q such that PQ can be chosen out of n is nC2 ways a general element of A must satisfy one of the following possibilities : (here general element be ai(1in))
(i) aiP and aiQ
(ii) aiP and aiQ
(iii) aiP and aiQ
(iv) aiP and aiQ
Let a1,a2PQ
there is only one choice each of them (i.e. (i) choice) and three choices (ii), (iii) and (iv) for each of remaining (n2) elements.
No. of ways of remaining elements =3n2
Hence, the no. of ways in which PQ contains exactly two elements = nC2×3n2

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