Question

A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing the elements of P. A subset Q is again chosen at random. The probability that P and Q are disjoint sets, is A. ( 1 2 )n B. ( 1 4 )n C. 3 4 D. ( 3 4 )n Select the correct answer from above options

Answer 1

Correct Answer - D The set A has n elements. So, it has 2 n 2 subsets. Therefore, set P can be chosen in 2 n C 1 2 ways. Similarly, set Q can also be chosen in 2 n C 1 2 ways. ∴ ∴ Sets P and Q can be chosen in . 2 n C 1 × . 2 n C 1 = 2 n × 2 n = 4 n . ways. Suppose P contains r elements, where r varies from 0 to n. Then, P can be chosen in . n C r . ways. For Q to be disjoint from A, it should be chosen from the set of all subsets of set consisting of remaining n-r elements. This can be done in 2 n − r 2 ways. Therefore, P and Q can be chosen in . n C r × 2 n − r . ways. But, r can vary from 0 to n. Therefore, the total number of ways of selecting P and Q such that they are disjoint is n ∑ r = 0 . n C r 2 n − r = ( 1 + 2 ) n = 3 n ∑ Hence, required probability = 3 n 4 n = ( 3 4 ) n =