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(B) The total number of ways to select two distinct numbers from $50$ is $^{50}C_2 = \frac{50 	imes 49}{2} = 1225$. For the product $ab$ to be divisi

Vedclass · Accepted Answer

$\frac{664}{1225}$

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(B) The total number of ways to select two distinct numbers from $50$ is $^{50}C_2 = \frac{50 	imes 49}{2} = 1225$. For the product $ab$ to be divisible by $3$, at least one of the numbers must be a multiple of $3$. It is easier to calculate the complement: the probability that the product $ab$ is $NOT$ divisible by $3$. This happens if neither $a$ nor $b$ is a multiple of $3$. In the set ${1, 2, \dots, 50}$, the multiples of $3$ are ${3, 6, \dots, 48}$, which are $16$ numbers. The numbers that are $NOT$ multiples of $3$ are $50 - 16 = 34$ numbers. The number of ways to choose two distinct numbers that are not multiples of $3$ is $^{34}C_2 = \frac{34 	imes 33}{2} = 17 	imes 33 = 561$. The probability that the product is not divisible by $3$ is $\frac{561}{1225}$. Therefore, the probability that the product $ab$ is divisible by $3$ is $1 - \frac{561}{1225} = \frac{1225 - 561}{1225} = \frac{664}{1225}$.

Two distinct numbers $a$ and $b$ are selected at random from ${1, 2, 3, \dots, 50}$. The probability that their product $ab$ is divisible by $3$ is:

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