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(A) The total number of subsets of a set $X$ containing $n$ elements is $2^n$.Since two subsets $A$ and $B$ are chosen,the total number of ways to cho

Vedclass · Accepted Answer

$\frac{^{2n}C_n}{2^{2n}}$

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(A) The total number of subsets of a set $X$ containing $n$ elements is $2^n$.Since two subsets $A$ and $B$ are chosen,the total number of ways to choose the pair $(A, B)$ is $2^n 	imes 2^n = 2^{2n}$.For $A$ and $B$ to have the same number of elements,let this number be $r$,where $0 \le r \le n$.The number of ways to choose a subset with $r$ elements is $^nC_r$.Thus,the number of ways to choose $A$ and $B$ such that $|A| = |B| = r$ is $(^nC_r)^2$.The total number of favorable outcomes is $\sum_{r=0}^{n} (^nC_r)^2$.Using the identity $\sum_{r=0}^{n} (^nC_r)^2 = ^{2n}C_n$,the total number of favorable ways is $^{2n}C_n$.Therefore,the required probability is $\frac{^{2n}C_n}{2^{2n}}$.

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:

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