{"status": "success", "data": {"description_md": "Six ants simultaneously stand on the six [[vertex|vertices]] of a regular [[octahedron]], with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex?\n\n$\\mathrm{(A)}\\ \\frac {5}{256}<br>\\qquad\\mathrm{(B)}\\ \\frac {21}{1024}<br>\\qquad\\mathrm{(C)}\\ \\frac {11}{512}<br>\\qquad\\mathrm{(D)}\\ \\frac {23}{1024}<br>\\qquad\\mathrm{(E)}\\ \\frac {3}{128}$\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>Six ants simultaneously stand on the six [[vertex|vertices]] of a regular [[octahedron]], with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ \\frac {5}{256}\\qquad\\mathrm{(B)}\\ \\frac {21}{1024}\\qquad\\mathrm{(C)}\\ \\frac {11}{512}\\qquad\\mathrm{(D)}\\ \\frac {23}{1024}\\qquad\\mathrm{(E)}\\ \\frac {3}{128}</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": "2005 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/05_amc12B_p24"}}