# How many permutations of the letters of the word 'MADHUBANI'

Question:

How many permutations of the letters of the word 'MADHUBANI' do not begin with M but end with I?

Solution:

Number of words that only end with $I=$ Number of permutations of the remaining 8 letters, taken all at a time $=\frac{8 !}{2 !}$

Number of words that start with $M$ and end with $I=$ Permutations of the remaining 7 letters, taken all at a time $=\frac{7 !}{2 !}$

Number of words that do not begin with M but end with I = Number of words that only end with I - Number of words that start with M and end with I

$=\frac{8 !}{2 !}-\frac{7 !}{2 !}$

$=17640$