JEE Main 2026 — 22 January, Morning Shift — Question 4

Two distinct numbers $a,b$ are chosen from $\{1,2,\dots,50\}.$

Total number of ways:

$$\binom{50}{2}=1225$$

The product $ab$ is divisible by $3$ if at least one of $a,b$ is divisible by $3$.

Count multiples of 3

$$\left\lfloor \frac{50}{3} \right\rfloor = 16$$

So, Multiples of $3 =16,\qquad$

Non-multiples of $3=50-16=34$

Count unfavorable cases

Unfavorable case: neither $a$ nor $b$ is divisible by 3.

$$\binom{34}{2}=\frac{34\cdot33}{2}=561$$

Favorable cases

Favorable = 1225-561=664

Probability

$$P=\frac{664}{1225}$$ $$\boxed{\frac{664}{1225}}$$