2020 Pascal Problem 23

Ali, Bea, Che, and Deb compete in a checkers tournament. Each player plays each other player exactly once. At the end of each game, either the two players tie or one player wins and the other player loses. A player earns 5 points for a win, 0 points for a loss, and 2 points for a tie. Exactly how many of the following final point distributions are possible?

\begin{array}{c|c} \text{Player} & \text{Points} \\ \hline \text{Ali} & 15 \\ \text{Bea} & 7 \\ \text{Che} & 4 \\ \text{Deb} & 2 \end{array}\qquad \begin{array}{c|c} \text{Player} & \text{Points} \\ \hline \text{Ali} & 10 \\ \text{Bea} & 10 \\ \text{Che} & 4 \\ \text{Deb} & 4 \end{array}\qquad \begin{array}{c|c} \text{Player} & \text{Points} \\ \hline \text{Ali} & 15 \\ \text{Bea} & 5 \\ \text{Che} & 5 \\ \text{Deb} & 2 \end{array}\qquad \begin{array}{c|c} \text{Player} & \text{Points} \\ \hline \text{Ali} & 12 \\ \text{Bea} & 10 \\ \text{Che} & 5 \\ \text{Deb} & 0 \end{array}

\textbf{(A)}\ 0\quad \textbf{(B)}\ 1\quad \textbf{(C)}\ 2\quad \textbf{(D)}\ 3\quad \textbf{(E)}\ 4

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 Pascal contests here.