= 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 $\lambda=N$. Thus
$$
\boxed{\widehat\theta_{ij}=\frac{n_{ij}}N.}
$$
Because every count is positive, the <log-likelihood> is strictly concave on the interior simplex and this solution is its unique maximum.
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