Question

How many distinguishable 11 letter​ "words" can be formed using the letters in MISSISSIPPI​?

Answer 1

This is given by the multinomial coefficient:

$\dbinom{11}{1,4,4,2}=\dfrac{11!}{1!4!4!2!}=34650$

If you're not familiar with the multinomial coefficient, you may be able to see it more clearly if you count the number of possible combinations taking each distinct letter $n$ times, where $n$ is the number of times it shows up in the original word.

$\underbrace{\dbinom{11}1}_{\text{M}}\underbrace{\dbinom{10}4}_{\text{I}}\underbrace{\dbinom64}_{\text{S}}\underbrace{\dbinom22}_{\text{P}}=\dfrac{11!}{1!10!}\dfrac{10!}{4!6!}\dfrac{6!}{4!2!}\dfrac{2!}{2!0!}=\dfrac{11!}{1!4!4!2!}$