{"status": "success", "data": {"description_md": "A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those 9 people is equal to the reciprocal of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people. How many values $n$, $9 \\leq n \\leq 2017$, can be the number of participants?\n\n$\\textbf{(A)}\\ 477 \\qquad \\textbf{(B)}\\ 482 \\qquad \\textbf{(C)}\\ 487 \\qquad \\textbf{(D)}\\ 557 \\qquad\\textbf{(E)}\\ 562$\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>A set of  <span class=\"katex--inline\">n</span>  people participate in an online video basketball tournament. Each person may be a member of any number of  <span class=\"katex--inline\">5</span> -player teams, but no teams may have exactly the same  <span class=\"katex--inline\">5</span>  members. The site statistics show a curious fact: The average, over all subsets of size  <span class=\"katex--inline\">9</span>  of the set of  <span class=\"katex--inline\">n</span>  participants, of the number of complete teams whose members are among those 9 people is equal to the reciprocal of the average, over all subsets of size  <span class=\"katex--inline\">8</span>  of the set of  <span class=\"katex--inline\">n</span>  participants, of the number of complete teams whose members are among those  <span class=\"katex--inline\">8</span>  people. How many values  <span class=\"katex--inline\">n</span> ,  <span class=\"katex--inline\">9 \\leq n \\leq 2017</span> , can be the number of participants?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 477 \\qquad \\textbf{(B)}\\ 482 \\qquad \\textbf{(C)}\\ 487 \\qquad \\textbf{(D)}\\ 557 \\qquad\\textbf{(E)}\\ 562</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": 5, "problem_name": "2017 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/17_amc12B_p24"}}