Your denominator of $26^{11}$ is correct, but your numerator should use the multinomial coefficient $\displaystyle \binom{11}{1,4,4,2} = \frac{11!}{1!4!4!2!}$.
Why? Well, the multinomial coefficient $\displaystyle \binom{n}{n_1, n_2, ..., n_k}$ tells us how many ways there are to put $n$ objects into $k$ groups of sizes $n_1, n_2, ..., n_k$, where the sizes add up to $n$. For example, we might have $10$ different objects and we want to figure out how many ways there are to distribute them into $3$ bins of sizes $3, 5, 2$. Then the formula tells us there are $\displaystyle \frac{10!}{3!5!2!} = 2,520$ ways.
With the word "MISSISSIPPI", you can think of there being $11$ spots, and we have to assign each of these spots to a letter -- one of M, I, S or P. There is 1 M, 4 I's, 4 S's, and 2 P's. So, we can think of this question as asking how many ways are there to take $11$ objects and putting them into four groups of sizes 1, 4, 4 and 2. This is simply the multinomial coefficient $\displaystyle \binom{11}{1,4,4,2} = \frac{11!}{1!4!4!2!} = 34,650$.
Edit: Alternatively, another way to think about it is, as another answer pointed out, as a permutation with repetitions. You have to divide by those factorials so as to remove repeats.
Hence, there are 11! ways of getting these specific letters.
<-- Why? No. $\endgroup$ – zipirovich May 26 '20 at 1:15