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Question

Let X be a set containing n elements. If two subsets A and B of X are picked at random, the probability that A and B have the same number of elements is

A
2nCn2n
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B
12nCn
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C
135(2n1)2nn!
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D
3n4n
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Solution

The correct option is A 2nCn2n
We know that the number of subsets of a set containing n elements is 2n.
Therefore, the number of ways of choosing A and B is 2n2n=22n.
We also know that the number of subsets (of X) which contain exactly r elements is nCr.
Therefore, the number of ways
of choosing A and B so that they have the same number of elements is
(nC0)2+(nC1)2+(nC2)2+....+(nCn)2
=2nCn=123(2n1)(2n)n!n!
=[135(2n1)][246(2n)]n!n!
=2n(n!)[135(2n1)]n!n!=2n[135(2n1)]n!
Thus, the probability of the required event is
=2nCn22n=135(2n1)2n(n!)

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