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 Q is a subset of P, is A. 34n B. (34)n C. n(34)n D. 3n4n Select the correct answer from above options

Answer 1

Correct Answer - B The set A has n elements. So, it has 2n subsets. Therefore, set P can be chosen in 2nC1 ways. Similarly, set Q can also be chosen in 2nC1 ways. ∴ Sets P and Q can be chosen in .2nC1×.2nC1=2n×2n=4n ways. Let the subset P of A contains r elements, with 0≤r≤n. Then, the number of ways of choosing P is .nCr. The subset Q of P can have at most r elements and the number of ways of choosing Q is 2r. Therefore, the number of ways of choosing P and Q is .nCr×2r when P has r elements. So, P and Q can be chosen in general in n ∑ r=0 .nCr×2r=(1+2)n=3n ways Hence, required probability = 3n 4n =( 3 4 )n