Holiday Contest 2025 - Team Round - Problem 6

Toshiro must convince a girl that he is orz at math. To do this, he must split five distinct people into groups of size two or more. Three of these people are Toshiro's friends and will say that Toshiro is orz. The other two hate Toshiro and will say that he is norz (not orz). Toshiro needs to split the people so that, in EVERY group, the people who say Toshiro is orz is greater than or equal to the people who say he is norz. In other words, if there are n people in a group, the people who say Toshiro is orz in that group should be m \geq \left\lceil{\frac{n}{2}}\right\rceil. How many ways can Toshiro achieve this?