Perms and Combs Question - Dealing with 'Mississippi'

Question

Firstly I apologise for my lack of formatting, I'm looking into fixing it.

Q: How many distinct arrangements of the letters in 'Mississippi' are there if,

(i) all four I's do not come together.

I had this as a complementary event, so all arrangements subtract the ones that have the I's together.

However, I calculated the I's together as 8!/4!. The answers say 8!/4!2!

Where did the 2! come from?

Once I find this I can subtract from all arrangements to find the solution.

Answer 1

How many distinct ways can we arrange letters of $MISSISSIPPI$ such that the $I$'s are all together. I think of it like this: the $I$'s are behaving as just one symbol, so count all the permutations on just 8 letters.

$$ M, S, S, S, S, P, P, IIII $$

This is $8!$. The letter $S$ is repeated 4 times so divide by $4!$ not to overcount the re-arrangements of $S$. And the letter $P$ is repeated 2 times so divide by $2!$.

Answer 2

The $2!$ is due to the two $p$'s in the word. The $8!/4!$ permutations that you've found treats the two $p$'s as distinct objects. But they're identical, so divide by $2!$