distinct permutations using all objects with repeated objects I thought my question was identical to Number of permutations with repeated objects. but it is not. They were looking for permutations of size $0\leq ... \leq N$ but I would not call a permutation of size less than $N$  a "permutation". 
I'm looking for the number of distinguishable distinct permutations of N objects where some of the objects are indistinguishable copies. Suppose there are $k$ total types of objects. To use that questions terminology: "In general, suppose we have objects $\underbrace{X_1, \dotsc, X_1}_{n_1}, \underbrace{X_2, \dotsc, X_2}_{n_2}, \dotsc,\dotsc, \dotsc, \underbrace{X_k, \dotsc, X_k}_{n_k}$. Then what is the number of ways we can choose and order $N=n_1 +\dotsb + n_k$? 
E.g., if the objects are "AAB" so N=3, then "AAB" and "AAB" are indistinguishable so the answer is 3 (AAB,ABA,BAA).
Is it a simple modification of the answer there?

Assume they were distinguishable, total would be $N!$.
Assuming they we not distinguishable as you state, each of the $N!$ can be mapped to $n_1! n_2! \dots n_k!$ permutations without the distinguishing factor (permuting the kinds among themselves)
Thus the answer is $$\frac{N!}{n_1! n_2! \dots n_k!}$$
