I am not sure if this is the right place to ask this question, but I recently came across this problem in a math competition and I have written a proof for this. However, I am not really sure if my proof is rigorous or even correct at all. That's why I am here since I need help to verify if it is correct and if I did not assume anything without proving it.
Here is the problem:
There are $30$ balls in a box. The balls are in $6$ colours. For each colour, there are $5$ balls. If we draw $5$ balls randomly and simultaneously, how many combinations are there about the colours of $5$ balls?
This is my solution:
