<s>[INST] The number of ways in which one can choose three distinct numbers from the set so that the product of the chosen numbers is divisible by $9$ 
Consider the set $A=\{1,2,3,...,30\}$
  . The number of ways in which one can choose three 
  distinct numbers from  $A$  so that the product of the chosen numbers is divisible by $9$  is $X$

Find $X$
This question is from KVPY SA 2017.
My working:
Nos in set divisible by $3$(but not by $9$)-$3,6,12,15,21,24,30$-Total $7$
Nos in set divisible by $9$-$9,18,27$-Total $3$
Therefore, the ans should be $7.7.20+7.7.7+7.7.3+7.3.3+3.3.3+3.20.20+3.3.20+3.7.20$
Where $.$ is multiplication
Is this correct?
 [/INST] You haven't adressed "three distinct numbers" correctly. For instance, you aren't allowed to pick $3, 3, 5$, so the first term shouldn't be $7\cdot 7\cdot 20$, it should be $7\cdot 6\cdot 20$. And then, still, you count $3, 6, 5$ as a separate choice from $6, 3, 5$. Therefore, what you really want is $\frac{7\cdot 6}2\cdot 20$ for that term.
With these two corrections, the final answer is
$$
\frac{7\cdot 6}{2}\cdot 20 + \frac{7\cdot 6\cdot 5}{6} + \frac{7\cdot 6}{2}\cdot 3 + 7\cdot\frac{3\cdot 2}{2} + \frac{3\cdot 2\cdot 1}6 + 3\cdot \frac{20\cdot 19}2 + \frac{3\cdot 2}2\cdot 20 + 3\cdot 7\cdot 20
$$
</s>