2023 Cayley Problem 10

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

\textbf{(A)}\ \frac{5}{16}\quad \textbf{(B)}\ \frac{13}{32}\quad \textbf{(C)}\ \frac{9}{25}\quad \textbf{(D)}\ \frac{1}{2}\quad \textbf{(E)}\ \frac{7}{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.