Number of ways to put $n$ objects with $k$ distinguishable groups into sets of size $a$, $b$, and $c$ $(a+b+c = n)$

Question

If I have $n$ objects in $k ≤ n$ distinguishable groups, for the case of $k = n$ (all objects distinguishable), I believe the number of ways to choose, without replacement, $a$ objects, then $b$ objects, then $c$ objects (where $a + b + c = n$) is $\frac{n!}{a!b!c!}$. I think in the case of $k = 1$ (all objects indistinguishable), the answer is just 1 (?). I'm wondering how this can be extended for $1 < k < n$? Does the solution change whether $a, b, c > 0$ or if one of them equals 0?