Given a population of unknown size, with k tagged individuals, we sample n of them and find m to be tagged. We would like an estimate for the size of the population N If we assume equal likelyhood of capture throughout and constant population size, we can apply the Lincoln-Petersen method and estimate N using the ratio: $kn/m$, I would like to formally derive this ratio by maximizing the discrete probability function of tagged indvidualsin the sample. i.e. find N such that the probability of having m tagged individuals in our sample of size n is maximized.
We know that P(m out of n individuals are tagged)=${k\choose m}{N-k\choose n-n}/{N\choose n}$. I would like to show that the ratio / is the value of N that maximizes this function but I/m getting stuck in the computation.