{"status": "success", "data": {"description_md": "Let $S$ be a set with six elements. Let $P$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, not necessarily distinct, are chosen independently and at random from $P$. the probability that $B$ is contained in at least one of $A$ or $S-A$ is $\\frac{m}{n^{r}},$ where $m$, $n$, and $r$ are positive integers, $n$ is prime, and $m$ and $n$ are relatively prime. Find $m+n+r.$ (The set $S-A$ is the set of all elements of $S$ which are not in $A.$)\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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 S be a set with six elements. Let P be the set of all subsets of S. Subsets A and B of S, not necessarily distinct, are chosen independently and at random from P. the probability that B is contained in at least one of A or S-A is \\frac{m}{n^{r}}, where m, n, and r are positive integers, n is prime, and m and n are relatively prime. Find m+n+r. (The set S-A is the set of all elements of S which are not in A.)

Leading zeroes must be inputted, so if your answer is 34, then input 034. 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": "2007 AIME II Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/07_aime_II_p11", "prev": "/problem/07_aime_II_p09"}}