 Solution

Under complete randomization, every assignment vector with exactly $n_z$ patients receiving treatment $z$ is equally likely. Thus, for $z=(z_1,\dots ,z_n)$ and $N_r(z)={\sum }_i{1}_{\{z_i=r\}}$,The reciprocal probability $n!/(n_0!n_1!n_2!)$ is the multinomial coefficient counting assignments with those arm sizes.