The question is : Number of Derangement possible for, "ABHIBHAV".
But my concern is not solving only typically this problem, but to solve any derangement problem.
I recently came to know that these problem can be easily solved with Rooks polynomial, which intrigued me alot coz as such these problem require a greater thinking skills and I was amazed to know how is rooks polynomial solving this problem.
Then I posted one question on this website only (which is this)
After reading that I found that such problem are solved this way: (Bcoz I absolutely could not get any "good","easy" AND "standard" articles which could make my understanding clear, yes those 3 words are with respect to me ..you may consider me as dumb)
So here is the solution I found :(Using rooks polynomial)
"ABHIBHAV" here I and V as single entity so $L1^2$ and there are three pair of letters A,B,H means $L2^3$
$L1^2 = (x-1)^2$
$L2^3 = (x^2-4x+2)^3$
So this is the reason behind the above $$L_n(x) = \sum_{k=0}^n (-1)^k { n \choose k}^2 k! x^{n-k},$$
How these terms come? (No one better can explain than this beautifully written answer to similar problem..)
Now to get the derangemnts I am actually suppose to first multiply those big polynomial equation so as to get another polynomial which ofcourse is too too too much work.. Then after multiplying those two things I get one more huge equation (here) an 8th power equation..
So calculation: $x^8 - 14 x^7 + 79 x^6 - 232 x^5 + 386 x^4 - 376 x^3 + 212 x^2 - 64 x + 8$ (9 terms)
Doesn't it seem to be too much multiplication. Is this the right way to solve these by this too lengthy polynomial multiplication method So the answer comes like this $$answer=8!-14.7!+79.6!-232.5!+386.4!-376.3!+212.2!-64.1!+8.0!=772$$
Now this is what I learned how I learned from The methods I read.
(The procedure I am telling is my interpretation of the answers on the questions I linked...bcoz I had mentioned that I didn't find any good and easy source so having confusion with the basics )
So my actually question is, is this all what rooks theorem is? Bcoz I have put up a recent question as a doubt which you find here where the answerer, gave me really a true insight of what actually is rooks theorem thus I realized that the whole rooks theorem is based on Chess board. So Now if I were asked a question to derange the letters of word "abc" (has nothing to do with "ABHIBHAV", just a simple demo) I would take a chess board of 3x3 dimension
So here in the chess board (1,1), (2,2),(3,3) are considered forbidden. (Bcoz Derangement) Now computing rooks polynomial and proceeding as per answer given here I get exactly the total number of derangements needed.
But when I proceed for the problem mentioned "ABHIBHAV".
Then how should the chess board be modelled to solve this problem. Bcoz here I have repitition of letters!
Any help of intuition to proceed / some extra links would be much appreciated!
(And one more thing: I am really missing some points and having a gap in my understanding of this concept and it's application..to solve question, so plz suggest any easy material (if possible) for further read..bcoz already I have the original research paper which is some how complex to read for me)
