what does this operation mean?

Question

What does $$ E\left( RI \right) =\frac{\sum_i{\left( \begin{array}{c} n_{i\cdot}\\ 2\\ \end{array} \right) \sum_j{\left( \begin{array}{c} n_{\cdot j}\\ 2\\ \end{array} \right)}}}{\left( \begin{array}{c} n\\ 2\\ \end{array} \right)} $$ mean?

Answer 1

It means the expectation of $RI$ is a product of two sums of binomial coefficients, divided by a binomial coefficient. In particular, $\binom{m}{2}=\frac12m(m-1)$ is the number of ways to choose $2$ distinct elements from $m$ if order is irrelevant.