{"status": "success", "data": {"description_md": "Each time a Clash Royale chest is opened, a card is randomly selected (the chance of each distinct card being selected is equal) from set $\\mathcal{S}$, which contains $N \\geq 2$ distinct cards, for some positive integer $N$. Then, a copy of the selected card is given to the player (Set $\\mathcal{S}$ remains unchanged, and there is an ample supply of each card). Given that the cards *\"Hog Rider\"* and *\"Ram Rider\"* are in $\\mathcal{S}$, and that the expected number of chests opened until a Hog Rider and a Ram Rider are drawn in that order is between $275$ and $300$, find the sum of the digits of $N$.\n\n$\\textbf{(A)}~7\\qquad\\textbf{(B)}~8\\qquad\\textbf{(C)}~10\\qquad\\textbf{(D)}~11\\qquad\\textbf{(E)}~14$", "description_html": "<p>Each time a Clash Royale chest is opened, a card is randomly selected (the chance of each distinct card being selected is equal) from set <span class=\"katex--inline\">\\mathcal{S}</span>, which contains <span class=\"katex--inline\">N \\geq 2</span> distinct cards, for some positive integer <span class=\"katex--inline\">N</span>. Then, a copy of the selected card is given to the player (Set <span class=\"katex--inline\">\\mathcal{S}</span> remains unchanged, and there is an ample supply of each card). Given that the cards <em>&#8220;Hog Rider&#8221;</em> and <em>&#8220;Ram Rider&#8221;</em> are in <span class=\"katex--inline\">\\mathcal{S}</span>, and that the expected number of chests opened until a Hog Rider and a Ram Rider are drawn in that order is between <span class=\"katex--inline\">275</span> and <span class=\"katex--inline\">300</span>, find the sum of the digits of <span class=\"katex--inline\">N</span>.</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~7\\qquad\\textbf{(B)}~8\\qquad\\textbf{(C)}~10\\qquad\\textbf{(D)}~11\\qquad\\textbf{(E)}~14</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2024 Mock AMC 12 - Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/2024_mock_amc12-p20", "prev": "/problem/2024_mock_amc12-p18"}}