Binomial

Let $S = \left\{0, 1, 2, ..., 8084\right\}$ . Then there are a total of $\left(\begin{array}{c}8085\\ 169\end{array}\right)$ ways to select $84+1+84=169$ distinct elements from $S$ .

Let $k$ be the median of the chosen 169 numbers. Then the range of the possible values of $k$ is the set $T = \left\{84, 85, ..., 8000\right\}$ . For each value of $k$ in $T$ , there are $\left(\begin{array}{c}k\\ 84\end{array}\right)$ ways to choose 84 numbers which are less than $k$ , and $\left(\begin{array}{c}8084-k\\ 84\end{array}\right)$ ways to choose 84 numbers more which are greater than $k$ , so that $k$ is the median. Therefore there are a total of $\left(\begin{array}{c}k\\ 84\end{array}\right)\left(\begin{array}{c}8084-k\\ 84\end{array}\right)$ ways to get the median $k$ .

Thus, $\left(\begin{array}{c}n\\ r\end{array}\right)=\left(\begin{array}{c}8085\\ 169\end{array}\right)$ , so the smallest possible value of $n+r$ is 8254.