2025 Gauss 8 Problem 25

The list 11, 12, 14, 23, 31, 44, 45, 46, 56, 64, 67, 74 can be arranged so that the units digit of each number matches the tens digit of the number that follows it. For example, 12, 23, 31, 11, 14, 44, 45, 56, 67, 74, 46, 64 is one such arrangement. How many such arrangements of the given list are possible?

\textbf{(A)}\ 18\quad \textbf{(B)}\ 24\quad \textbf{(C)}\ 36\quad \textbf{(D)}\ 30\quad \textbf{(E)}\ 12

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 Gauss contests here.