2023 Pascal Problem 14

Each of four doors is randomly either open or closed. What is the probability that exactly two of the four doors are open?

\textbf{(A)}\ \frac{3}{8}\quad \textbf{(B)}\ \frac{5}{16}\quad \textbf{(C)}\ \frac{1}{4}\quad \textbf{(D)}\ \frac{1}{2}\quad \textbf{(E)}\ \frac{3}{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 Pascal contests here.