I have this problem here:
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?
I know that Chelsea has a 1/6 chance of winning, and that Mel has 1/3 chance of winning. If I do 1/2^3 * 1/3^2 * 1/6, I get $\frac{1}{432}$. However, that only gives me the probability of the 3 players winning in that exact order. How can I find out the number of ways in which Alex, Mel, and Chelsea can win 3, 2, and 1 rounds, respectively