{"status": "success", "data": {"description_md": "Let $K$ be the number of sequences $A_1$, $A_2$, $\\ldots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\\{1, 2, 3, \\ldots, 10\\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\\{\\}$, $\\{5, 7\\}$, $\\{2, 5, 7\\}$, $\\{2, 5, 7\\}$, $\\{2, 5, 6, 7, 9\\}$ is one such sequence, with $n = 5$.What is the remainder when $K$ is divided by $10$?\n\n$\\textbf{(A) } 1 \\qquad \\textbf{(B) } 3 \\qquad \\textbf{(C) } 5 \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 9$\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": "

Let K be the number of sequences A_1 , A_2 , \\ldots , A_n such that n is a positive integer less than or equal to 10 , each A_i is a subset of \\{1, 2, 3, \\ldots, 10\\} , and A_{i-1} is a subset of A_i for each i between 2 and n , inclusive. For example, \\{\\} , \\{5, 7\\} , \\{2, 5, 7\\} , \\{2, 5, 7\\} , \\{2, 5, 6, 7, 9\\} is one such sequence, with n = 5 .What is the remainder when K is divided by 10 ?

\\textbf{(A) } 1 \\qquad \\textbf{(B) } 3 \\qquad \\textbf{(C) } 5 \\qquad \\textbf{(D) } 7 \\qquad \\textbf{(E) } 9

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": "2023 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p25", "prev": "/problem/23_amc12A_p23"}}