I have the following problem:
There is a set of $30$ elements. It must be split in $3$ subsets. However, each subset must have, at least, $9$ elements. How can I count how many ways are there to choose them?
I tried to solve it using a sum of multinomial coefficients with the possible combinations of the elements, $(9,9,12)$, $(9,10,11)$ and $(10,10,10)$ as
$\begin{equation} \binom{30}{9,9,12} + \binom{30}{9,10,11} + \binom{30}{10,10,10} \end{equation}$,
but I am not sure of it.