Given, the word MADHUBANI. It has 9 letters of which 1 letter (A) is repeating twice and all other letters of the word is are distinct.

To find: Number of ways the letters of word MADHUBANI be arranged in such a way that the word must not begin with M but ends with I.

Let's assume that I will be at the end of the word we are forming, and will not be changed its position. We will not take it into consideration now as its position is fixed. Now we have 8 letters to arrange.

First we find all arrangements of word MADHUBANI and then we minus all those arrangements of word MADHUBANI in such a way that the word is starting with the letter M. This will exactly be same as- all number of arrangements such that the word will not begins with the letters M and ends with the letter I (As the letter I was already fixed in the last position).

Since we know, Permutation of n objects taking r at a time is ⁿP_r,and permutation of n objects taking all at a time is n!

And, we also know Permutation of n objects taking all at a time having p objects of the same type, q objects of another type, r objects of another type is . i.e. the, number of repeated objects of same type are in denominator multiplication with factorial.

A total number of arrangements of word MADHUBANI excluding I: Total letters 8. Repeating letter A, repeating twice. The total number of arrangements will be equals to

Now we find a total number of arrangements such that the word begins with the letter M.

The total number of arrangements of word MADHUBANI excluding I will be equals to permutation of 7 objects (A, D, H, U, B, A, N) taking all together.

Letters : 7

Repeating letter: A (2 times)

Total number of word arranging all the letters

Now, a Total number of arrangements where the word not starts with M but ends with I (I was already fixed in the last position) will be equals to total arrangements of word MADHUBAN minus the total number of arrangements in such a way that word starts with letter M.

= 17640

Hence, a total number of arrangements of word MADHUBANI in such a way that the word is not starting with M but ends with I equal to 17640.