{"status": "success", "data": {"description_md": "**POTD December 5, 2023**\n**Author:** munch\n\nLet $\\pi = a_1, a_2, \\cdots, a_{8}$ be some permutation of the numbers $1, 2, \\cdots, n$. We say that two elements $a_i$ and $a_j$ of $\\pi$ form an inversion if $i < j$ and $a_i > a_j$. (For example, there are five inversions in the permutation $4312$; they are $31, 32, 41, 42,$ and $43$). Find the last $3$ digits of the total number of all permutations $\\pi$ of the numbers $1, 2, \\cdots, 8$ such that the number of inversions in $\\pi$ is a multiple of $6$.", "description_html": "<p><strong>POTD December 5, 2023</strong><br/>&#10;<strong>Author:</strong> munch</p>&#10;<p>Let <span class=\"katex--inline\">\\pi = a_1, a_2, \\cdots, a_{8}</span> be some permutation of the numbers <span class=\"katex--inline\">1, 2, \\cdots, n</span>. We say that two elements <span class=\"katex--inline\">a_i</span> and <span class=\"katex--inline\">a_j</span> of <span class=\"katex--inline\">\\pi</span> form an inversion if <span class=\"katex--inline\">i &lt; j</span> and <span class=\"katex--inline\">a_i &gt; a_j</span>. (For example, there are five inversions in the permutation <span class=\"katex--inline\">4312</span>; they are <span class=\"katex--inline\">31, 32, 41, 42,</span> and <span class=\"katex--inline\">43</span>). Find the last <span class=\"katex--inline\">3</span> digits of the total number of all permutations <span class=\"katex--inline\">\\pi</span> of the numbers <span class=\"katex--inline\">1, 2, \\cdots, 8</span> such that the number of inversions in <span class=\"katex--inline\">\\pi</span> is a multiple of <span class=\"katex--inline\">6</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "Problem of the Day #16", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}