{"status": "success", "data": {"description_md": "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?\n\n$\\textbf{(A)}\\ \\frac{5}{72}\\qquad\\textbf{(B)}\\ \\frac{5}{36}\\qquad\\textbf{(C)}\\ \\frac{1}{6}\\qquad\\textbf{(D)}\\ \\frac{1}{3}\\qquad\\textbf{(E)}\\ 1$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Alex, Mel, and Chelsea play a game that has  <span class=\"katex--inline\">6</span>  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  <span class=\"katex--inline\">\\frac{1}{2}</span> , 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?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{5}{72}\\qquad\\textbf{(B)}\\ \\frac{5}{36}\\qquad\\textbf{(C)}\\ \\frac{1}{6}\\qquad\\textbf{(D)}\\ \\frac{1}{3}\\qquad\\textbf{(E)}\\ 1</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2012 AMC 12A Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12A_p12", "prev": "/problem/12_amc12A_p10"}}