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 ...
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Piyush Sawarkar
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1You could use the Inclusion-Exclusion Principle. – N. F. Taussig Feb 23 '20 at 20:18
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Link, If you tell your class (if you are in) then I can help you with the content because there are many documents that have high standard mathematics. – Nikola Alfredi Feb 28 '20 at 10:05
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Another Link – Nikola Alfredi Feb 28 '20 at 10:12
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@NikolaAlfredi basically I am just an engg undergrad..but still I'd rather like If you can share some content...which you were talking about...!! – Piyush Sawarkar Mar 01 '20 at 19:08
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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 $$\displaystyle l_n(x) = \sum_{k = 0} ^n (-1)^k {n \choose k}^2 k!\ x^{n - k} $$
Nikola Alfredi
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LINK, I am in 10th grade and it was difficult for me to understand it. I still need to understand it accurately, I just know how to use it. – Nikola Alfredi Mar 02 '20 at 17:00
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Hi! Can you tell me if we can apply this formula in cases where only some of the letters are needed to be deranged. For example: in questions like this How many permutations of 1,... 8 are there in which no even number appears in its natural position? Now in the case which I asked earlier there every letter was supposed to be deranged but here it's not like this, only even have to be deranged and odd have no restriction!, Yes I know this could be solved by principle of inclusion and exclusion but how to use the formula you gave here? – Piyush Sawarkar Jun 23 '20 at 16:19
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or also in case of question like find derangemnt of "BHI BHV" where BHV have to be deranged and BHI have no restriction. – Piyush Sawarkar Jun 23 '20 at 16:20
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if you want I may even edit the question a bit so that you may add an answer. – Piyush Sawarkar Jun 23 '20 at 16:30
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You can ask another question... send me the link. And sorry for the delayed reply. – Nikola Alfredi Jul 01 '20 at 09:53
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Thanks @Nikola Alfredi for responding, but actually I had already asked it on this site and also got the answer for the same and have understood what is basically that rooks polynomial! – Piyush Sawarkar Jul 01 '20 at 19:23