{"status": "success", "data": {"description_md": "Forty slips are placed into a hat, each bearing a number $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, or $10$, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b \\neq a$. What is the value of $\\frac{q}{p}$?\n\n$\\mathrm{(A)} 162 \\qquad \\mathrm{(B)} 180 \\qquad \\mathrm{(C)} 324 \\qquad \\mathrm{(D)} 360 \\qquad \\mathrm{(E)} 720$", "description_html": "<p>Forty slips are placed into a hat, each bearing a number  <span class=\"katex--inline\">1</span> ,  <span class=\"katex--inline\">2</span> ,  <span class=\"katex--inline\">3</span> ,  <span class=\"katex--inline\">4</span> ,  <span class=\"katex--inline\">5</span> ,  <span class=\"katex--inline\">6</span> ,  <span class=\"katex--inline\">7</span> ,  <span class=\"katex--inline\">8</span> ,  <span class=\"katex--inline\">9</span> , or  <span class=\"katex--inline\">10</span> , with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let  <span class=\"katex--inline\">p</span>  be the probability that all four slips bear the same number. Let  <span class=\"katex--inline\">q</span>  be the probability that two of the slips bear a number  <span class=\"katex--inline\">a</span>  and the other two bear a number  <span class=\"katex--inline\">b \\neq a</span> . What is the value of  <span class=\"katex--inline\">\\frac{q}{p}</span> ?</p>\n<p> <span class=\"katex--inline\">\\mathrm{(A)} 162 \\qquad \\mathrm{(B)} 180 \\qquad \\mathrm{(C)} 324 \\qquad \\mathrm{(D)} 360 \\qquad \\mathrm{(E)} 720</span> </p>\n<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": 4, "problem_name": "2005 AMC 10B Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc10B_p22", "prev": "/problem/05_amc10B_p20"}}