Solution
= Solution
The three stage counts are independent Poisson variables of mean $\rho$ under the product law. Hence
$$
\boxed{P(N=n)=e^{-3\rho}\frac{(3\rho)^n}{n!}.}
$$
Dividing the joint law by this mass gives
$$
\boxed{P(C=c,R=r,B=b\mid N=n)
=\frac{n!}{c!r!b!}\left(\frac13\right)^n,\quad c+r+b=n.}
$$
Thus the conditional stage composition is multinomial with equal stage probabilities.