{"status": "success", "data": {"description_md": "Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\\frac pq,$ where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3,17,$ and $10.$) What is $p+q?$\n\n$\\textbf{(A) }55 \\qquad \\textbf{(B) }56 \\qquad \\textbf{(C) }57\\qquad \\textbf{(D) }58 \\qquad \\textbf{(E) }59$\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": "

Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin i is 2^{-i} for i=1,2,3,.... More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is \\frac pq, where p and q are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins 3,17, and 10. ) What is p+q?

\\textbf{(A) }55 \\qquad \\textbf{(B) }56 \\qquad \\textbf{(C) }57\\qquad \\textbf{(D) }58 \\qquad \\textbf{(E) }59

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2021 AMC 12B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12B_p24", "prev": "/problem/21_amc12B_p22"}}