Is $1_1,1_2,1_3$ different from $1_2,1_3,1_1$ in a permutation?

Question

This is from problem 1 in Introduction to Probability by Joseph K Blitzstein and Jessica Hwang.

1. How many ways are there to permute the letters in the word MISSISSIPPI?

My answer was 11!. I'm reading that it should be $11! \over 4!4!2!$. My understanding is that a permutation is unique as long as the order of the values are separate. eg: $1_1,1_2,1_3$ is different from $1_3,1_1,1_2$

Is that not correct?

Answer 1

Your understanding is correct; $1_1 1_2 1_3$ would be different from $1_3 1_1 1_2$ because there the symbols are labeled. However, the letters in MISSISSIPPI are not labeled. The word

MISSISSIPPI

is indistinguishable from the word

MISSISSIPPI

even though I've secretly switched the first two S's in the second word. The two $S$'s are not distinguishable.

Here we are considering permutations of a word, which may have repeatedly letters, as opposed to a permutation of a set, which has distinct elements. The definitions are different.