In the box there are $91$ balls, where $m$ are red, and the rest are blue. To estimate unknown parameter $m$, at once $19$ balls are drawn, $7$ being red and $12$ being blue. Based on the given sample, estimate $m$ using method of maximum likelihood.
I'm aware this is the hypergeometric distribution in question. I'm having trouble finding the maximum using MLE and how would I use these $19$ drawn balls to get the answer?
You need to find the $m\in\{7,\ldots,79\}$ that maximizes $$ \frac{\binom{m}{7}\binom{91-m}{19-7}}{\binom{91}{19}}. $$ The fastest way I guess would be to calculate the 73 options with some code. The result you should get is the following.
The maximum is attained for $m=27$.