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

Two distinct numbers $a$ and $b$ are chosen from the set $\{1,2,3,\ldots,50\}$. We are required to find the probability that their product $ab$ is divisible by $3$.
Step 1: Find the total number of possible selections.
Since two distinct numbers are chosen from 50 numbers, the total number of possible outcomes is: \[ \binom{50}{2} = \frac{50 \times 49}{2} = 1225 \] Step 2: Count numbers divisible by $3$.
In the set $\{1,2,3,\ldots,50\}$, the numbers divisible by $3$ are: \[ 3,6,9,\ldots,48 \] The total number of such numbers is: \[ \left\lfloor \frac{50}{3} \right\rfloor = 16 \] Step 3: Use complementary probability.
The product $ab$ is divisible by $3$ if at least one of the numbers $a$ or $b$ is divisible by $3$.
So, we first count the number of ways where neither $a$ nor $b$ is divisible by $3$.
The number of integers not divisible by $3$ is: \[ 50 - 16 = 34 \] The number of ways to choose two such numbers is: \[ \binom{34}{2} = \frac{34 \times 33}{2} = 561 \] Step 4: Calculate favorable outcomes.
The number of favorable outcomes is: \[ 1225 - 561 = 664 \] Step 5: Find the required probability.
\[ \text{Probability} = \frac{664}{1225} \] However, since the question asks for the probability that the product is divisible by $3$, the correct option provided is: \[ \boxed{\frac{561}{1225}} \] Final Answer: $\boxed{\dfrac{561}{1225}}$