Question

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∩B contains exactly 10 elements is Select the correct answer from above options

Answer 1

Correct Answer - C::D Since, set A contains n elements n elements. So, it has 2n subsets. ∴ Set P can be chosen in 2n ways, similarly set Q can be chosen in 2n ways. ∴ P and Q can be chosen in (2n)(2n)=4n ways. Suppose, P contains r elements, where varies from 0 to n. Then, P can be chosen in 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 2n-r ways. ∴ P and Q can be chosen in n Cr.2n-r ways. But, r can vary from 0 to n. ∴Totalνmberofdisj∮setsPandQ=overset(n)underset(r=0)sumoverset(n)""C_(r )2^(n-2)=(1+2)^(n)=3^(n)Hence,requiredprobability=(3^(n))/(4^(n))=((3)/(4))^(n)`