Q32 / 75 · JEE Main 2026 (22 January Shift 1) · Mathematics
Two distinct numbers $a$ and $b$ are selected at random from $1,2,3, \ldots, 50$. The probability, that their product $a b$ is divisible by 3, is
- A$\frac{272}{1225}$
- B$\frac{561}{1225}$
- Correct answerC$\frac{664}{1225}$
- D$\frac{8}{25}$
Solution
Total ways $= \binom{50}{2} = 1225$.
$ab$ is divisible by 3 iff at least one of $a, b$ is divisible by 3.
Multiples of 3 in $\{1,\ldots,50\}$: $3,6,\ldots,48 \Rightarrow 16$ numbers. Non-multiples of 3: $34$.
$P(\text{neither divisible by 3}) = \frac{\binom{34}{2}}{\binom{50}{2}} = \frac{561}{1225}$.
$P(ab \text{ divisible by 3}) = 1 - \frac{561}{1225} = \frac{664}{1225}$.
$ab$ is divisible by 3 iff at least one of $a, b$ is divisible by 3.
Multiples of 3 in $\{1,\ldots,50\}$: $3,6,\ldots,48 \Rightarrow 16$ numbers. Non-multiples of 3: $34$.
$P(\text{neither divisible by 3}) = \frac{\binom{34}{2}}{\binom{50}{2}} = \frac{561}{1225}$.
$P(ab \text{ divisible by 3}) = 1 - \frac{561}{1225} = \frac{664}{1225}$.
Hence, the correct answer is option: 'C'.