JEE Mathematics 2026 Question with Solution

Given: Two distinct numbers $a$ and $b$ are chosen from $\{1,2,3,\ldots,50\}$.

Find: The probability that $ab$ is divisible by $3$.

The total number of ways to choose two distinct numbers is

Numbers divisible by $3$ in $\{1,2,3,\ldots,50\}$ are $3,6,9,\ldots,48$, so their count is

Hence, the count of numbers not divisible by $3$ is

Use the complementary event: $ab$ is not divisible by $3$ only when neither $a$ nor $b$ is divisible by $3$.

So the number of such selections is

Therefore, the number of favorable selections is

So the required probability is

The solution lists option B and final answer $\dfrac{561}{1225}$, but the working clearly gives $\dfrac{664}{1225}$. Therefore, the correct option from the given choices is C.

Given: Two distinct numbers are selected from $1$ to $50$.

Find: Probability that the product is divisible by $3$.

Instead of directly counting cases where at least one number is divisible by $3$, count the opposite case first.

There are $16$ multiples of $3$ up to $50$, so there are $34$ numbers not divisible by $3$. The product fails to be divisible by $3$ only if both chosen numbers come from these $34$ numbers.

Thus,

Hence,

Therefore, the correct option is C.