Exact maximum likelihood estimator of the structure of a stratified population

An exact expression for the extreme values of the integer vector $\overline N=(N_1,\dots,N_k)$ that maximize the function
$$\prod_{j=1}^k\binom{N_j}{l_j}$$
for arbitrary integers $l_1>0,\dots,l_k>0$, $k\ge2$, and a given $N^0=N_1+\dots+N_k$ is derived. Also, statistical applications of the result are discussed.