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(D) The total number of ways to choose two distinct numbers $x$ and $y$ from the set $\{1, 2, \dots, 15\}$ is $15 	imes 14 = 210$.The equation of the

Vedclass · Accepted Answer

$\frac{1}{42}$

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(D) The total number of ways to choose two distinct numbers $x$ and $y$ from the set $\{1, 2, \dots, 15\}$ is $15 	imes 14 = 210$.The equation of the line passing through $(0, 0)$ with a slope of $\frac{2}{3}$ is $y = \frac{2}{3}x$,which simplifies to $2x = 3y$.We need to find pairs $(x, y)$ such that $x, y \in \{1, 2, \dots, 15\}$ and $2x = 3y$. This implies $x$ must be a multiple of $3$. Let $x = 3k$,then $2(3k) = 3y$,so $y = 2k$.For $k=1, x=3, y=2$.For $k=2, x=6, y=4$.For $k=3, x=9, y=6$.For $k=4, x=12, y=8$.For $k=5, x=15, y=10$.There are $5$ such favorable pairs.The probability is $\frac{5}{210} = \frac{1}{42}$.

Two numbers $x$ and $y$ are chosen at random from the set of integers $\{1, 2, 3, 4, \dots, 15\}$. The probability that the point $(x, y)$ lies on a line passing through $(0, 0)$ with a slope of $\frac{2}{3}$ is:

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