{"status": "success", "data": {"description_md": "**POTD January 22, 2024**\n\nLet $f(S)$ be the number of pairs of consecutive numbers in the set $S$. For example,\n\n$[f \\left(\\{\\mathbf{1}, \\mathbf{2}, 5, 7, \\mathbf{9}, \\mathbf{10}, 12\\} \\right) = 2.$\n\nFor a random subset $T \\subseteq \\{1,2,\\ldots,2024\\}$ such that $|T| = k$, the expected value of $f(T)$ is $E(k)$. Find the number of factors of $\\sum_{i=0}^{2024} E(i)$.", "description_html": "<p><strong>POTD January 22, 2024</strong></p>&#10;<p>Let <span class=\"katex--inline\">f(S)</span> be the number of pairs of consecutive numbers in the set <span class=\"katex--inline\">S</span>. For example,</p>&#10;<p><span class=\"katex--inline\">[f \\left(\\{\\mathbf{1}, \\mathbf{2}, 5, 7, \\mathbf{9}, \\mathbf{10}, 12\\} \\right) = 2.</span></p>&#10;<p>For a random subset <span class=\"katex--inline\">T \\subseteq \\{1,2,\\ldots,2024\\}</span> such that <span class=\"katex--inline\">|T| = k</span>, the expected value of <span class=\"katex--inline\">f(T)</span> is <span class=\"katex--inline\">E(k)</span>. Find the number of factors of <span class=\"katex--inline\">\\sum_{i=0}^{2024} E(i)</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "Problem of the Day #48", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}