Derangement with letters repeated.

Question

Find the total number of Derangement of the word: "mississippi". Can some one please suggest a concrete method in how to deal with Derangement with repeated letters.. I solved a question like Derangement of "Bottle", using help of an answer on stack exchange(you can see it here), but I was unable to comprehend when more than 1 letters got repeated which is the case with "mississippi".. As per the answer to find derangement for Bottle I did: $$(D6-D4-2D5)/2$$ So please suggest some concrete methods and also some references for further reading ...

Answer 1

SOLUTION :

$$\displaystyle \frac {1}{4! \ .4! \ .2!} \int_0 ^{\infty} e^{-x} (x - 1) (x^2 - 4x + 2)(x^4 - 16x^3 + 72x^2 - 96x + 24)^2 dx = 648 $$

Rook's Polynomial

Where $\displaystyle l_2(x) = x^2 - 4x + 2 $

And $\displaystyle l_4(x) = x^4 - 16x^3 + 72x^2 - 96x + 24 $

And $$\displaystyle l_n(x) = \sum_{k = 0} ^n (-1)^k {n \choose k}^2 k!\ x^{n - k} $$