Question

Let (X be a set containing n elements. If two subsets A and B of X are picked at random, the probability that A and B have the same number of elements, is

• A.
• B.
• C.
• D.

Answer 1

Answer: (A)

Answer 2

We know that the number of sub-sets of set containing n elements is 2ⁿ. Therefore the number of ways of choosing A and B is 2ⁿ. 2ⁿ = 2²ⁿ.

We also know that the number of sub-sets (of X) which contain exactly r elements is ⁿC_r. Therefore the number of ways of choosing A and B, so that they have the same number elements is

$\left( { } ^ { n } C _ { o } \right) ^ { 2 } + \left( { } ^ { n } C _ { 1 } \right) ^ { 2 } + \left( { } ^ { n } C _ { 2 } \right) ^ { 2 } + \ldots . \left( { } ^ { n } C _ { n } \right) ^ { 2 } = { } ^ { 2 n } C _ { n }$

Thus the required probability = .