\begin{gather*} E[X(X-1)] = \sum_{x=0}^n x(x-1) \frac{\binom{n_1}{x} \binom{n-n_1}{r-x}}{\binom{n}{r}}\\ = \sum_{x=2}^n x(x-1) \frac{\frac{n_1!}{x(x-1)(x-2)!(n_1-x)! } \binom{n-n_1}{r-x}}{\binom{n}{r}}\\ = \sum_{x=2}^n \frac{\frac{n_1!}{(x-2)!(n_1-x)! } \frac{n_2!}{(r-x)!(n_2-r+x)!}}{\binom{n}{r}}\\ = n_1 \cdot (n_1-1) \cdot \sum_{x=2}^n \frac{\frac{(n_3)!}{(x-2)!(n_3 -(x-2))! } \frac{n_2!}{((r-2)-(x-2))!(n_2-(r-2)+(x-2))!}}{\binom{n}{r}}\\ = n_1 \cdot (n_1-1) \sum_{y=0}^{m} \frac{\frac{(n_3)!}{y!(n_3 -y)! } \frac{n_2!}{(s-y)!(n_2-s+y)!}}{\binom{n}{r}}\\ = \frac{n_1 \cdot (n_1-1) \cdot r \cdot (r-1)}{n (n-1)} \sum_{y=0}^{m} \frac{\binom{(n_3)}{y} \binom{n_2}{s-y}}{\binom{m}{s}}\\ = \frac{n_1 \cdot (n_1-1) \cdot r \cdot (r-1)}{n (n-1)} \end{gather*}