Gambling Trio II

Alice, Bob, and Chad all go to the casino to gamble. They start with </code>a, $b,$ and </code>c$ , respectively, where $a,b,c > 0$ are integers. The dealer selects two of three players at each round to play against each other. Alice, Bob, and Chad are equally skilled at gambling, so all rounds are fair. The loser pays $$1$ to the winner. In this scenario, the gamblers can go into debt and continue gambling. Find the expected number of rounds played until the sum of the pairwise products of their fortunes is $$0$ when $a = 5, b = 10,$ and $c = 15$ .