Question 77
Three schools $A$, $B$ and $C$ participate in a chess tournament with five students from each school. Let $a_1, a_2, \dots, a_5; b_1, b_2, \dots, b_5; c_1, c_2, \dots, c_5$ be the list of the players from $A$, $B$, $C$ respectively. Let $P_A$, $P_B$, $P_C$ be the scores that schools $A$, $B$, $C$ make, respectively, when the tournament is over following the rules described below. Find the remainder of the number of possible triples $(P_A, P_B, P_C)$ when it is divided by $8$.
1. Players from each school have matches in order from the first one, and if a player is once defeated then he (or she) is eliminated from the tournament. The first match of the tournament is between the players $a_1$ and $b_1$.
2. When $x_i$ from school $X$ is defeated by $y_i$ from $Y$, if there are players left from the school $Z$ other than $X$, $Y$ then $y_i$ have another match with a player from $Z$, if there is no player left in $Z$ but there are players from $X$ then $y_j$ have a match with $x_{i+1}$. The tournament is over when no player is left in two schools.
3. School $X$ adds $10^{i-1}$ points to its score when $x_i$ from $X$ wins a match.
1. Players from each school have matches in order from the first one, and if a player is once defeated then he (or she) is eliminated from the tournament. The first match of the tournament is between the players $a_1$ and $b_1$.
2. When $x_i$ from school $X$ is defeated by $y_i$ from $Y$, if there are players left from the school $Z$ other than $X$, $Y$ then $y_i$ have another match with a player from $Z$, if there is no player left in $Z$ but there are players from $X$ then $y_j$ have a match with $x_{i+1}$. The tournament is over when no player is left in two schools.
3. School $X$ adds $10^{i-1}$ points to its score when $x_i$ from $X$ wins a match.