Question

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

• A. $\dfrac{8}{25}$
• B. $\dfrac{561}{1225}$
• C. $\dfrac{664}{1225}$
• D. $\dfrac{272}{1225}$

Hint:

For divisibility-based probability problems, it is often easier to use the complementary event and subtract from the total number of outcomes.

Answer 1

The Correct Option is B

To solve this problem, we need to determine the probability that the product of two randomly selected distinct numbers from the set \(\{1, 2, 3, \ldots, 50\}\) is divisible by 3. Let's break it down step by step:

1. Identify the number of ways to select two distinct numbers from the given set:

The total number of ways to select two distinct numbers from 50 numbers is given by the combination formula: \(C(50, 2) = \frac{50 \times 49}{2} = 1225\)

1. Identify the integers in the set divisible by 3:

The numbers divisible by 3 up to 50 are \(\{3, 6, 9, \ldots, 48\}\). This forms an arithmetic sequence with:

• First term (\(a\)) = 3
• Common difference (\(d\)) = 3
• Last term = 48

Using the formula for the number of terms in an arithmetic sequence: \(n = \frac{48 - 3}{3} + 1 = 16\) Thus, there are 16 numbers that are divisible by 3.

1. Calculate the number of ways to select two numbers such that their product is not divisible by 3. To achieve this, neither number should be divisible by 3:

The total number of numbers not divisible by 3 from 1 to 50 is 50 - 16 = 34. The number of ways to choose two numbers from these 34 numbers is: \(C(34, 2) = \frac{34 \times 33}{2} = 561\)

1. Calculate the probability that ab is not divisible by 3:

\[ P(\text{Product not divisible by 3}) = \frac{561}{1225} \]

1. Calculate the probability that the product ab is divisible by 3:

Since the complement of the event (product not divisible by 3) has a probability of \(\frac{561}{1225}\), the probability that the product is divisible by 3 is: \(1 - \frac{561}{1225} = \frac{664}{1225}\)

After reviewing the options, the probability that the product of two randomly chosen numbers is divisible by 3 is \(\frac{664}{1225}\). Thus, the correct answer given the correct option should actually be corrected in the source or verified for further detail.