AMC 12 2012a Problem 11

Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?

\textbf{(A)}\ \frac{5}{72} \textbf{(B)}\ \frac{5}{36} \textbf{(C)}\ \frac{1}{6} \textbf{(D)}\ \frac{1}{3} \textbf{(E)}\ 1