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 of A is again chosen at random. Find the probability that P and Q have no common elements.

Answer 1

Since, set A contains r elements. So, it has 2ⁿ subsets. 

.'. Set P can be chosen in 2ⁿ ways, similarly set Q can be chosen 2ⁿ ways. 

.'. P and Q can be chosen in (2ⁿ)(2ⁿ) = 4ⁿways.

Suppose, P contains r elements, where r varies from 0 to n.

Then, P can be chosen in ⁿC_r ways, for 0 to be disjoint from A, it should be chosen from the set of ali subsets of set consisting of remaining (n - r)elements. This can be done in 2^{n - r} ways. 

.'. P and Q can be chosen in ⁿC_r . 2^{n - r} ways. 

But, P can vary from 0 to n. 

.'. Total number of disjoint sets P and Q