Past exam of the options II course of the University of Oxford / 2013 / O5 / a

Ignoring a factor independent of the probabilities, the likelihood function is $L(\pi )={\prod }_i{\pi }_i^{n_i}$, so its log is $\ell (\pi )={\sum }_i{n}_i\log {\pi }_i$ subject to ${\sum }_i{\pi }_i=1$. A Lagrange multiplier gives $n_i/{\pi }_i=\lambda$. Summing ${\pi }_i=n_i/\lambda$ makes $\lambda =n$.

ThusSince all counts are positive, a zero fitted cell probability would have zero likelihood. On the interior the log-likelihood is strictly concave, so this stationary point is its unique global maximum.