Two numbers a and b are chosen at random from | vedclass

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

Question

(D) The total number of ways to choose two distinct numbers $a$ and $b$ from the set ${1, 2, 3, \dots, 30}$ is given by ${}^{30}C_2 = \frac{30 	imes

Vedclass · Accepted Answer

$\frac{47}{87}$

Vedclass · Answer

(D) The total number of ways to choose two distinct numbers $a$ and $b$ from the set ${1, 2, 3, \dots, 30}$ is given by ${}^{30}C_2 = \frac{30 	imes 29}{2} = 435$.We know that $a^2 - b^2$ is divisible by $3$ if $a^2 \equiv b^2 \pmod{3}$.For any integer $n$, $n^2 \pmod{3}$ can be $0$ (if $n$ is a multiple of $3$) or $1$ (if $n$ is not a multiple of $3$).Let $S_0$ be the set of numbers divisible by $3$ $(|S_0| = 10)$ and $S_1$ be the set of numbers not divisible by $3$ $(|S_1| = 20)$.$a^2 - b^2$ is divisible by $3$ in two cases:Case $1$: Both $a$ and $b$ are in $S_0$. The number of ways is ${}^{10}C_2 = \frac{10 	imes 9}{2} = 45$.Case $2$: Both $a$ and $b$ are in $S_1$. The number of ways is ${}^{20}C_2 = \frac{20 	imes 19}{2} = 190$.Total favorable cases $= 45 + 190 = 235$.Therefore, the required probability $= \frac{235}{435} = \frac{47}{87}$.

Two numbers $a$ and $b$ are chosen at random from the set of first $30$ natural numbers. The probability that $a^2 - b^2$ is divisible by $3$ is

Similar Questions

In a single throw of $2$ dice,find the probability of getting a total of $8.$

From $21$ tickets numbered $1, 2, 3, \ldots, 21$,one ticket is drawn at random. Find the probability that the ticket drawn has a number divisible by $3$.

$A$ man and his wife appear for an interview for two posts. The probability of the husband's selection is $\frac{1}{7}$ and that of the wife's selection is $\frac{1}{5}$. What is the probability that only one of them will be selected?

$A$ problem of mathematics is given to three students whose chances of solving the problem are $\frac{1}{3}$,$\frac{1}{4}$,and $\frac{1}{5}$ respectively. The probability that the question will be solved is

$A$ committee of $5$ members is to be formed from $6$ men and $4$ women. The probability that at least one woman is present in the committee is

Vedclass Test Series

Exam Paper Generator

Online Exam Module