Binomial coefficient to approach multi-way choices DP problem?
The binomial coefficient here appears through the formula
$$
\sum_{i=1}^{n-1} i = \binom{n}{2}.
$$
In the formula, $OPT(j)$ is the minimal cost of a solution for the first $j$ servers. Another way of looking it that it's the minimal cost up to $j$ of a solution which puts the file in $S_j$.

Where does the recurrence come form? Take an optimal solution for the first $i$ servers. We know that the file appears in $S_j$. Let $S_i$ be the next largest server it appears on (or $i=0$ if it only appears in $S_j$). Without loss of generality, we can assume that the solution restricted to $S_1,\ldots,S_i$ is optimal, and so of cost $OPT(i)$. Putting the file in $S_j$ costs $c_j$. Accessing the file at positions $i+1,\ldots,j-1$ costs $(j-(i+1)) + \cdots + (j-(j-1)) = \binom{j-i}{2}$. This gives the recurrence formula. (We also have to verify the case $i=0$ separately, which I leave to you.)