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Set ${\eta }_{ik}={\beta }_k^T{\mathbf{x}}_i$, so ${\eta }_{i1}=0$. The multinomial logistic regression implies $p_{ik}=e^{{\eta }_{ik}}{p}_{i1}$. Normalizing givesMultiply the multinomial likelihoods for conditionally independent groups. The multinomial coefficients are constant in $\beta$, leavingThe reference constraint ${\beta }_1=0$ identifies the coefficients: a common shift of every category coefficient otherwise leaves all probabilities unchanged. The PDF places the full sum inside the numerator exponent; the TeX breaks that expression.

This is not a valid ordinary R glm family specification. A multinomial distribution is natural for the three damage counts conditional on the row total, but base R's generalized linear model interface does not supply a family named multinomial. Furthermore, the written response is one cell count, not a three-category response or grouped multinomial vector. A properly specified multinomial logistic regression could model category probabilities by group, but it is not the model fit by this command. The independent-count log-linear model in part (a), conditional on the margins, is the appropriate supplied alternative.