# If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary,

**Question:**

If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the word 'MOTHER'.

**Solution:**

In a dictionary, the words are listed and ranked in alphabetical order. In the given problem, we need to find the rank of the word MOTHER.

For finding the number of words starting with E, we have to find the number of arrangements of the remaining 5 letters.

Number of such arrangements = 5!

For finding the number of words starting with H, we have to find the number of arrangements of the remaining 4 letters.

Number of such arrangements = 5!

For finding the number of words starting with M, fixing the next letter as E, we have to find the number of arrangements of the remaining 4 letters, which is 4!.

For finding the number of words starting with M, fixing the next letter as H, we have to find the number of arrangements of the remaining 4 letters, which is 4!.

For finding the number of words starting with M, fixing the second letter as O, and the third letter as E, we have to find the number of arrangements of the remaining 3 letters, which is 3!.

For finding the number of words starting with M, fixing the second letter as O, and the third letter as H, we have to find the number of arrangements of the remaining 3 letters, which is 3!.

For finding the number of words starting with M, fixing the second letter as O, and the third letter as R, we have to find the number of arrangements of the remaining 3 letters, which is 3!.

For finding the number of words starting with M, fixing the second letter as O, the third letter as T, and the fourth letter as E, we have to find the number of arrangements of the remaining 2 letters, which is 2!.

Now, the next word formed would be MOTHER.

Number of words after which we reach the word MOTHER = 5!+5!+4!+4!+3!+3!+3!+2!+1 = 309