Number of permutations with repeated objects.

Number of permutations with repeated objects.



I will use letters as objects. 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 $N$ objects $0 \leq N \leq n_1

+\dotsb + n_k$, i.e. the number of permutations? If $n_1 = \dotsb = n_k =

1$, then of course this is just a standard permutation problem. I am just

curious if there is any formula for it.

Note: I initially asked about combinations, which was pointed out to be a

duplicate of the question here. I have deleted the original post to ask

this question, instead.