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 cap Q` contain just one element, is

Options

A. `2^(2n)-.^(2n)C_(n)`
B. `2^(n)`
C. `2^(n)-1`
D. `3^(n)`

Answer

D

Explanation

Let `A={a_(1),a_(2),a_(3), . . .,a_(n)}`
(i) `a_(i) in P,a_(i) in Q`
(ii) `a_(i) in P, a _(1) cancel(in)Q`
(ii) `a_(i) cancel(in)P,a_(i) in Q`
(iv) `a_(i) cancel(in)P,a_(i)cancel(in)Q,` where `1 le I le n`
`because P cap Q=phi` [cases in favour 3 i.e., (ii), (iii),(iv)]
`therefore`Required number of ways`=3^(n)`.