P is a set containing n elements . A subset A of P is chosen and the set P is reconstructed by replacing the element of A. B such that A and B have no common elements, is

Options

A. `2^(n)`
B. `3^(n)`
C. `4^(n)`
D. none of these

Answer

b

Explanation

Suppose A contains ` r (0 le r len)` elements. Then , B
is constructed by selecting some elements from the remaining
`(n-r)` elements. Clearly, A can be chosen in `""^(n)C_(r)` ways and B in
`(""^(n-r)C_(0)+""^(n-r)C_(1)+...+""^(n-r)C_(n-r) )= 2^(n-r)` ways.So, total number
of ways of choosing A and B is `""^(n)C_(r) xx2^(n-r)`. But , r can very from
0 to n.
`therefore ` Required number of ways = `sum_(r=0)^(n) ""^(n)C_(r) xx2^(n-r) = (1 + 2)^(n) = 3^(n)` .