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.
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.
