Let A be a set consisting of n elements. The probability of selecting two subsets P and Q of set A such that `Q= overline(P)`, is

Question

Let A be a set consisting of n elements. The probability of selecting two subsets P and Q of set A such that Q= ˉ P , is A. 1 2 B. 1 2n-1 C. 1 2n D. 1 3n Select the correct answer from above options

Answer 1

Correct Answer - B The set A has 2n elements. Therefore, two subsets P and Q can be chosen in 2nC2 ways. Suppose P consists of r elements. Then, P can be chosen in .nCr . Since, Q= ˉ P . Therefore, P and Q can be chosen in .nCr ways. But, r can vary from 0 to n and P and Q can be interchanged also. ∴ Number of ways of selecting P and Q such that Q= ˉ P is 1 2 n ∑ r=0 .nCr=2n-1 Hence, required probability = 2n-1 .2nC2 = 1 2n-1