Problem: "How many distinguishable 11-letter 'words' can be formed using the letters in
MISSISSIPPI?"

Because "words" is in quotations, I interpret that to simply mean 11-letter permutations and not real language words.

While each permutation must contain 11 letters, only 4 of those are unique: M (exactly 1 occurrence), I (exactly 4 occurrences), S (exactly 4 occurrences), and P (exactly 2 occurrences).

11! cannot be the answer because that would produce many duplicate permutations (whereas the problem calls only for distinguishable ones). So the answer must be 11!-x where x is the number of duplicates.

Here I get stuck. Any help will be appreciated.

Google is well aware of this!
Example:
https://math.stackexchange.com/quest...ers-in-mississ

Originally Posted by NicholasDeMaio
Problem: "How many distinguishable 11-letter 'words' can be formed using the letters in
MISSISSIPPI?"
Because "words" is in quotations, I interpret that to simply mean 11-letter permutations and not real language words.
While each permutation must contain 11 letters, only 4 of those are unique: M (exactly 1 occurrence), I (exactly 4 occurrences), S (exactly 4 occurrences), and P (exactly 2 occurrences).
Consider these two 'word' $MISSISSIPPI$ and $MI_1S_1S_2I_2S_3S_4I_3P_1P_2I_4$
I hope that you agree those are indeed different letter strings.
And these are three different arangements:
$MI_1S_4S_2I_2S_1S_3I_3P_1P_2I_4$, $MI_1S_2S_3I_2S_4S_1I_3P_1P_2I_4$, $MI_1S_4S_24_2S_3S_1I_3P_1P_2I_4$ (places are the same but S-subscripts different)

To explain the point consider the string $"O_1UTDO_2O_3RS"$ made up of eight different characters.
We can rearrange that string in $8!$ ways.
How many ways rearrange the subscripted O's leaving all other letters fixed? Is it $3!~?$ WHY?
In the list of the eight factorial rearrangements with subscripts, lets go through and delete all subscripts.
Now in that list we would have $3!=6$ identical strings without subscripts in fixed positions.
So we need count only one out of six.
Thus, there are $\dfrac{8!}{3!}$ ways to rearrange the word 'OUTDOORS'
There are $\dfrac{11!}{(4!)^2(2!)}$ ways to rearrange the word 'MISSISSIPPI'