2021 Cayley Problem 23

A special six-sided die has its faces numbered 1 through 6 and has the property that rolling each number x is x times as likely as rolling a 1. For example, the probability of rolling a 5 is 5 times the probability of rolling a 1, while the probability of rolling a 2 is 2 times the probability of rolling a 1. Robbie and Francine play a game where they each roll this die three times, and the total of their three rolls is their score. The winner is the player with the highest score; if the two players are tied, neither player wins. After two rolls each, Robbie has a score of 8 and Francine has a score of 10. The probability that Robbie will win can be written in lowest terms as \frac{r}{400+s}, where r and s are positive integers. What is value of r+s?

\textbf{(A)}\ 96\quad \textbf{(B)}\ 86\quad \textbf{(C)}\ 76\quad \textbf{(D)}\ 66\quad \textbf{(E)}\ 56

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.