2024 SSMO Accuracy Round Problems/Problem 9
Problem
In the game of 
- set, there are 
unique cards, each card containing a 12 traits of variants 
and 
A full house in the game of Sset is defined to be a hand of cards in which for each trait, all cards in the hand either share the same variant or have all different variants. The number of hands that can be considered a full house in the game of set can be expressed as 
where 
and 
are positive integers and 
is minimized. Find 
Solution
Note that all full houses can contain at most 
cards. Now, 2 cards uniquely determine a hand. So, there are 
full house hands of 2 cards. As there are 6 permutations for 3 card hands, there are 
full house hands of 3 cards. Lastly, note that all one-card hands are considered full houses. Thus, the answer is ![\[\frac{3^{24}-3^{12}}{2}+\frac{3^{24}-3^{12}}{6}+3^{12} = \frac{2}{3}(3^{24})+\frac{1}{3}(3^{12}) =\]](http://latex.artofproblemsolving.com/1/a/4/1a4a9cce6e23ccc82c0f3519648c179bd7ff9e1a.png)
![\[3^{23}+3^{23}+3^{11}\implies 3+23+3+23+3+11 = \boxed{66}.\]](http://latex.artofproblemsolving.com/3/3/9/339a90add2cb8311ee8b549e9e87f8ec5634f5db.png)
~SMO_Team
