Let a random distribution $\mathcal{P}$ on the real line $\mathbb{R}$ have the mixture of Dirichlet processes. Let $S^{(n)}=(S_{1},\cdots,S_{n})$ be the random partition of the positive integer $n$ based on a sample of size $n$ from $\mathcal{P}$. For the number $K_{n}=S_{1}+\cdots+S_{n}$ of distinct observations among the sample, Yamato (2012) gives the asymptotic distribution of $K_{n}$ and the rate $O(1/\log^{1/3}n)$ of its convergence. In this pager we give the Edgeworth expansion for $K_{n}$ with the rate $O(1/\log^{2/5}n)$ and the rate $O(1/\log^{3/7}n)$.