How many ways can $10$ different kids be distributed to $4$ distinct classes? I want to find the number ways to distribute 10 different kids to 4 different classes, such that each class has at least 2 kids.
My attempt: First distribute 8 kids to 4 classes such that each class has exactly 2 kids. Then there are 16 option to split the remaining 2 kids.
This is my result: $\binom{10}{2}\binom{8}{2}\binom{6}{2}\binom{4}{2}*16$ which is wrong. I don't have exact answer but I know it's either: $226800, 75600, 44100$.
Thanks.

Note that there are 2 different size types for classes, $3,3,2,2$ or $4,2,2,2$. For the first one, choose firstly which classes will have $3$ kids, $\binom{4}{2}$, then distribute the kids, $\binom{10}{3} \cdot \binom{7}{3} \cdot \binom{4}{2} \cdot \binom{2}{2}$. Do the same for the other size type and you will arrive at $ \binom{4}{2} \cdot \binom{10}{3} \cdot \binom{7}{3} \cdot \binom{4}{2} \cdot \binom{2}{2} + \binom{4}{1} \cdot \binom{10}{4} \cdot \binom{6}{2} \cdot \binom{4}{2} \cdot \binom{2}{2} = 226800$
