2018 Cayley Problem 24

There are more than 1\,000\,000 ways in which n identical black socks and 2n identical gold socks can be arranged in a row so that there are at least 2 gold socks between any 2 black socks. The sum of the digits of the smallest possible value of n is

\textbf{(A)}\ 9\quad \textbf{(B)}\ 10\quad \textbf{(C)}\ 11\quad \textbf{(D)}\ 12\quad \textbf{(E)}\ 13

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.