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 the same number of elements, is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

We know that the number of sub-sets of a set containing n elements is $2^n$.
Therefore the number of ways of choosing A and B is $2^n.2^n=2^{2n}$
We also know that the number of sub-sets (of X) which contain exactly r elements is ${}^n{C}_r.$
Therefore the number of ways of choosing A and B, so that they have the same number elements is
${(}^n{C}_0{)}^2+{(}^n{C}_1{)}^2+{(}^n{C}_2{)}^2+....+{(}^n{C}_n{)}^2{=}^{2n}{C}_n$
Thus the required probability = $\frac{{}^{2n}{C}_n}{2^{2n}}.$