Two distinct numbers a and b are selected at random from 1,2,3,,50.

Question

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

Accepted Answer

Total ways $=\binom{50}{2}=1225$.

$ab$ is not divisible by 3 only if neither $a$ nor $b$ is divisible by 3. Multiples of 3 in $1$ to $50$: $\lfloor 50/3\rfloor=16$, so numbers not divisible by 3: $50-16=34$.

Unfavourable pairs $=\binom{34}{2}=\dfrac{34\times33}{2}=561$.

Favourable pairs $=1225-561=664$.

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

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