Let X be a set containing n elements. If two | vedclass

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Question

(A) The total number of subsets of a set $X$ with $n$ elements is $2^n$. Since $A$ and $B$ are chosen from these subsets,the total number of ways to c

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) The total number of subsets of a set $X$ with $n$ elements is $2^n$. Since $A$ and $B$ are chosen from these subsets,the total number of ways to choose the pair $(A, B)$ is $2^n 	imes 2^n = 2^{2n}$. The number of subsets of $X$ having $r$ elements is given by ${}^nC_r$. For $A$ and $B$ to have the same number of elements,say $r$,we must choose $A$ with $r$ elements and $B$ with $r$ elements. The number of ways to choose $A$ and $B$ such that $|A| = |B| = r$ is ${}^nC_r 	imes {}^nC_r = ({}^nC_r)^2$. Summing over all possible values of $r$ from $0$ to $n$,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 favorable outcomes is ${}^{2n}C_n$. Thus,the 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 chosen at random,what is the probability that $A$ and $B$ have the same number of elements?

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