{"status": "success", "data": {"description_md": "Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\\frac{1}{12}$. What is the least possible number of faces on the two dice combined?\n\n$\\textbf{(A) }16 \\qquad \\textbf{(B) }17 \\qquad \\textbf{(C) }18\\qquad \\textbf{(D) }19 \\qquad \\textbf{(E) }20$\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>Two fair dice, each with at least  <span class=\"katex--inline\">6</span>  faces are rolled. On each face of each die is printed a distinct integer from  <span class=\"katex--inline\">1</span>  to the number of faces on that die, inclusive. The probability of rolling a sum of  <span class=\"katex--inline\">7</span>  is  <span class=\"katex--inline\">\\frac34</span>  of the probability of rolling a sum of  <span class=\"katex--inline\">10,</span>  and the probability of rolling a sum of  <span class=\"katex--inline\">12</span>  is  <span class=\"katex--inline\">\\frac{1}{12}</span> . What is the least possible number of faces on the two dice combined?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }16 \\qquad \\textbf{(B) }17 \\qquad \\textbf{(C) }18\\qquad \\textbf{(D) }19 \\qquad \\textbf{(E) }20</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": "2021 AMC 12B Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12B_p20", "prev": "/problem/21_amc12B_p18"}}