I've found that if there are $N$ objects split up into two groups of $n_1$, and $n_2=N-n_1$ then this can be divided into $$\frac{N!}{n_1!(N-n_1)!}$$ groups.
How can I extend this onto splitting $N$ objects split into three groups of $n_1, n_2, n_3$ if we assume that the objects are distinguishable, where $n_2=N-n_1$ and $n_3=N-n_1-n_2$?