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

The product ab will be divisibe by 3 only if one of a or b is divisible by 3. 

An easier approach to solving this would be to find the complement and then subtract it from 1. We count the complement: probability that $$ ab$$ is not divisible by 3, i.e., neither $$ a$$ nor $$ b$$ is divisible by 3.

From 1 to 50, Multiples of 3 = $$ \lfloor 50/3 \rfloor = 16 $$ . Hence, numbers not divisible by 3 = 50 - 16 = 34.

Now the total number of ways to choose 2 distinct numbers is $$ \binom{50}{2} $$

And the number of ways to select 2 numbers where both are not divisible by 3 is $$ \binom{34}{2} $$

So, $$ P(\text{not divisible by 3}) = \dfrac{\binom{34}{2}}{\binom{50}{2}} = \dfrac{34 \cdot 33}{50 \cdot 49} $$

Hence, $$ P(\text{divisible by 3}) = 1 - \dfrac{34 \cdot 33}{50 \cdot 49} = 1 - \dfrac{1122}{2450}= \dfrac{2450 - 1122}{2450} = \dfrac{1328}{2450} = \dfrac{664}{1225} $$