2016 Cayley Problem 10

Five students play chess matches against each other. Each student plays three matches against each of the other students. How many matches are played in total?

\textbf{(A)}\ 15\quad \textbf{(B)}\ 8\quad \textbf{(C)}\ 30\quad \textbf{(D)}\ 60\quad \textbf{(E)}\ 16

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.