{"status": "success", "data": {"description_md": "How many $15$-letter arrangements of $5$ A's, $5$ B's, and $5$ C's have no A's in the first $5$ letters, no B's in the next $5$ letters, and no C's in the last $5$ letters?\n\n$\\textrm{(A)}\\ \\sum\\limits_{k=0}^{5}\\binom{5}{k}^{3}\\qquad\\textrm{(B)}\\ 3^{5}\\cdot 2^{5}\\qquad\\textrm{(C)}\\ 2^{15}\\qquad\\textrm{(D)}\\ \\frac{15!}{(5!)^{3}}\\qquad\\textrm{(E)}\\ 3^{15}$\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 <span class=\"katex--inline\">15</span>-letter arrangements of <span class=\"katex--inline\">5</span> A&#8217;s, <span class=\"katex--inline\">5</span> B&#8217;s, and <span class=\"katex--inline\">5</span> C&#8217;s have no A&#8217;s in the first <span class=\"katex--inline\">5</span> letters, no B&#8217;s in the next <span class=\"katex--inline\">5</span> letters, and no C&#8217;s in the last <span class=\"katex--inline\">5</span> letters?</p>&#10;<p><span class=\"katex--inline\">\\textrm{(A)}\\ \\sum\\limits_{k=0}^{5}\\binom{5}{k}^{3}\\qquad\\textrm{(B)}\\ 3^{5}\\cdot 2^{5}\\qquad\\textrm{(C)}\\ 2^{15}\\qquad\\textrm{(D)}\\ \\frac{15!}{(5!)^{3}}\\qquad\\textrm{(E)}\\ 3^{15}</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2003 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/03_amc12A_p21", "prev": "/problem/03_amc12A_p19"}}