how many permutations can be formed from n objects when m of them are indistinguishable?

Question

Alright so if i have

n! Is the number of permutations assuming all objects are distinguishable

and we want m of the n objects indistinguishable so I think I would do something of the form

$n_1$ indistinguishable objects of type 1

$n_2$ indistinguishable objects of type 2

...

$n_k$ indistinguishable objects of type k

but since the total of indistinguishable is supposed to be m

m = $n_1$ + $n_2$ + ... + $n_k$

I think, but I'm not sure what $n_1$ etc.. would be

maybe just

n!/m!

Answer 1

It is advantageous to consider a more general problem. Assume there are $k$ types of objects, each kind $i$ having $n_i$ indistinguishable representatives, the overall number of objects being $n=\sum_{i=1}^k n_i$.

Then the overall number of possible permutations is determined by the multinomial coefficient: $$ \frac{n!}{\prod_{i=1}^k n_i!}. $$

Observe that this expression remains valid also when some $n_i$ are $0$.