 Solution

Ignoring the multinomial coefficient, the likelihood is $L(\theta )={\prod }_{i,j}{\theta }_{ij}^{n_{ij}}$. Maximize $\ell ={\sum }_{i,j}{n}_{ij}\log {\theta }_{ij}$ with $\sum {\theta }_{ij}=1$. A Lagrange multiplier gives $n_{ij}/{\theta }_{ij}=\lambda$; summing the resulting ${\theta }_{ij}=n_{ij}/\lambda$ gives $\lambda =N$. ThusBecause every count is positive, the log-likelihood is strictly concave on the interior simplex and this solution is its unique maximum.