A subset A of the set `X = {1.2,3.....100}` is chosen at random. The set X is reconstructed by replacing the elements of A, and another subset B of X is chosen at random. Then the probability that `A nn B` contains exactly 10 elements is

Answer

C, D

Explanation

Since, set A contains n elements n elements. So, it has `2^(n)` subsets.
`:.` Set P can be chosen in `2^(n)` ways, similarly set Q can be chosen in `2^(n)` ways.
`:. ` P and Q can be chosen in `(2^(n))(2^(n))=4^(n)` ways.
Suppose, P contains r elements, where varies from 0 to n. Then, P can be chosen in `overset(n)C_(r )` ways, for 0 to be disjoint from A, it should be chosen from the set of all subsets of set consisting of remainning (n-r) elements. This can be done in `2^(n-r)` ways.
`:.` P and Q can be chosen in `overset(n)""C_(r ).2^(n-r)` ways.
But, r can vary from 0 to n.
`:. Total number of disjoint sets P and Q
`=overset(n)underset(r=0)sumoverset(n)""C_(r )2^(n-2)=(1+2)^(n)=3^(n)`
Hence, required probability`=(3^(n))/(4^(n))=((3)/(4))^(n)`