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∪Q contain just one element, is A. 3 4n B. 3n 4n C. n( 3 4 )n D. n 4n Select the correct answer from above options

Answer 1

Correct Answer - B 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. If P ∪ Q P contains exactly one element, P and Q each can have at most one element. That is, if P has no element, Q must have one element, and the number of ways of choosing P and Q is . n C 0 × . n C 1 = n . . On the other hand, if P has one element, Q can be empty or it can be equal to P i.e. Q can be chosen in two ways, so that the number of ways of choosing P and Q is . n C 1 × 2 = 2 n . . Therefore, the number of ways of choosing P and Q in this case is n+2n=3n. Hence, the required probability = 3 n 4 n =