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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

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?

Similar Questions

$A, B, C, D$ cut a pack of $52$ well-shuffled playing cards successively in the same order. If the person who cuts a spade first wins the game and the game continues until this happens, then the probability that $A$ wins the game is

If the events $A$ and $B$ are mutually exclusive events such that $P(A) = \frac{3x + 1}{3}$ and $P(B) = \frac{1 - x}{4}$,then the set of possible values of $x$ lies in the interval

$A$ bag contains $5$ distinct Red,$4$ distinct Green,and $3$ distinct Black balls. If balls are drawn one by one without replacement,what is the probability of getting a particular red ball in the fourth draw?

One card is selected at random from $27$ cards numbered from $1$ to $27$. What is the probability that the number on the card is even or divisible by $5$?

Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined: $P(A) = 0.5$,$P(B) = 0.7$,$P(A \cap B) = 0.6$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module