Rao-Blackwellization of a multinomial probability product ($\widehat{p_0{p}_1{p}_2}=n_0{n}_1{n}_2/[n(n-1)(n-2)]$)

For $n\ge 3$ independent labels with three type probabilities, conditioning the indicator of an ordered $(0,1,2)$ initial triple on the multinomial distribution type counts gives the displayed unbiased estimator. The Rao-Blackwell theorem reduces its variance strictly when all three probabilities are positive. At a boundary parameter with a missing type, both variances vanish.