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Question

A is a set containing n different 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 choosing P and Q so that PQ contains exactly two elements is:

A
nC3×2n
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B
nC2×3n2
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C
3n2
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D
None of these
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Solution

The correct option is B nC2×3n2
Set A has n elements
Also, subset P and subset Q have exactly 2 common elements.
It means if we remove those two elements, then subset P and Q will have no common elements.
Ways of selecting 2 elements from n elements =nC2
Let subset P contains a elements where 0an2.
Then subset B must be chosen from (n2a) elements.
Suppose subset P contains 1element,thenpossiblewaysofselectingsubsetPwillbe{}^{ n-1 }{ C }_{1}$
And possible ways of selecting subset Q from remaining n3 elements =2n3
So total possibility if subset P contains 1 element =n2C1×2n3
Similarly for a=0,a=2,a=3...............a=n2
Combining all situation nC2×(n2C0×2n2+n2C1×2n3...............n2Cn2×20)=nC2×(1+2)n2=nC2×3n2

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