{"status": "success", "data": {"description_md": "Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$\n\n$\\textbf{(A)} ~47 \\qquad\\textbf{(B)} ~94 \\qquad\\textbf{(C)} ~227 \\qquad\\textbf{(D)} ~471 \\qquad\\textbf{(E)} ~542$", "description_html": "<p>Ang, Ben, and Jasmin each have  <span class=\"katex--inline\">5</span>  blocks, colored red, blue, yellow, white, and green; and there are  <span class=\"katex--inline\">5</span>  empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives  <span class=\"katex--inline\">3</span>  blocks all of the same color is  <span class=\"katex--inline\">\\frac{m}{n}</span> , where  <span class=\"katex--inline\">m</span>  and  <span class=\"katex--inline\">n</span>  are relatively prime positive integers. What is  <span class=\"katex--inline\">m + n ?</span> </p>\n<p> <span class=\"katex--inline\">\\textbf{(A)} ~47 \\qquad\\textbf{(B)} ~94 \\qquad\\textbf{(C)} ~227 \\qquad\\textbf{(D)} ~471 \\qquad\\textbf{(E)} ~542</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": "2021 AMC 10B Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc10B_p23", "prev": "/problem/21_amc10B_p21"}}