= Solution
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 $\lambda=n$.
Thus
$$
\boxed{\widehat\pi_i=n_i/n.}
$$
Since 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.
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