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.
$$\frac{{{}^{2n}{C_n}}}{{{2^{2n}}}}$$
B.
$$\frac{1}{{{}^{2n}{C_n}}}$$
C.
$$\frac{{1 \cdot 3 \cdot 5.....\left( {2n + 1} \right)}}{{{2^n}n!}}$$
D.
$$\frac{{{3^n}}}{{{4^n}}}$$
Answer :
$$\frac{{{}^{2n}{C_n}}}{{{2^{2n}}}}$$
Solution :
$$\eqalign{
& {\text{Required probability :}} \cr
& = \frac{{\sum\limits_{r = 0}^n {{}^n{C_r} \times {}^n{C_r}} }}{{{2^n} \times {2^n}}} \cr
& = \frac{{C_0^2 + C_1^2 + C_2^2 + ..... + C_n^2}}{{{4^n}}} \cr
& = \frac{{{}^{2n}{C_n}}}{{{2^{2n}}}} \cr} $$