{"status": "success", "data": {"description_md": "How many non-empty subsets $S$ of $\\lbrace 1,2,3,\\ldots ,15\\rbrace$ have the following two properties? \n\n$(1)$  No two consecutive integers belong to $S$.\n\n$(2)$  If $S$ contains $k$ elements, then $S$ contains no number less than $k$.\n\n$\\mathrm{(A) \\ } 277\\qquad \\mathrm{(B) \\ } 311\\qquad \\mathrm{(C) \\ } 376\\qquad \\mathrm{(D) \\ } 377\\qquad \\mathrm{(E) \\ }  405$\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>How many non-empty subsets  <span class=\"katex--inline\">S</span>  of  <span class=\"katex--inline\">\\lbrace 1,2,3,\\ldots ,15\\rbrace</span>  have the following two properties?</p>&#10;<p> <span class=\"katex--inline\">(1)</span>   No two consecutive integers belong to  <span class=\"katex--inline\">S</span> .</p>&#10;<p> <span class=\"katex--inline\">(2)</span>   If  <span class=\"katex--inline\">S</span>  contains  <span class=\"katex--inline\">k</span>  elements, then  <span class=\"katex--inline\">S</span>  contains no number less than  <span class=\"katex--inline\">k</span> .</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } 277\\qquad \\mathrm{(B) \\ } 311\\qquad \\mathrm{(C) \\ } 376\\qquad \\mathrm{(D) \\ } 377\\qquad \\mathrm{(E) \\ }  405</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": "2006 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/06_amc12A_p24"}}