2020 Cayley Problem 23

Ricardo wants to arrange three 1s, three 2s, two 3s, and one 4 to form nine-digit positive integers with the properties that

- when reading from left to right, there is at least one 1 before the first 2, at least one 2 before the first 3, and at least one 3 before the 4, and

- no digit 2 can be next to another 2.

(For example, the integer 121\,321\,234 satisfies these properties.) In total, how many such nine-digit positive integers can Ricardo make?

\textbf{(A)}\ 278\quad \textbf{(B)}\ 260\quad \textbf{(C)}\ 254\quad \textbf{(D)}\ 272\quad \textbf{(E)}\ 266

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.