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'.

In the dictionary the words at each stage are arranged in alphabetical order. In the given problem we must therefore consider the words beginning with E, H, M, 0, R, T in order. E will occur in the first place as often as there are ways of arranging the remaining 5 letters

$\therefore$ Number of words starting with E = 5!

= $5\times 4\times 3\times 2\times 1=120$

Number of words starting with H 5!

=120

Number of words beginning with M is 5!, but one of these words is the word MOTHER.

So, we first find the number of words beginning with ME and MH.

Number of words starting with ME = 4! = $4\times 3\times 2\times 1$ = 24

Now, the words beginning with 'MO' must follow.

There are 4! words beginning with MO, one of these words is the word Mother itself.

So, we first find the number of words beginning with MOE, MOH and MOR.

Number of words starting with MOH= 3! = 6

Number of words starting with MOR= 3! = 6

Number of words starting with MOE= 3! = 6

Number of words beginning with MOT is 3! but one of these words is the word MOTHER itself

So, we first find the number of words beginning with MOTE.

Number of words starting with MOTE = 2! = 2

Now, the words beginning with MOTH must follow.

There are 2! words beginning with MOTH, one of these words is word MOTHER itself.

The first word beginning with MOTH is the word MOTHER.

\therefore Rank of Mother

= $2\times 120+2\times 24+3\times 6+2+1$

= 240 + 48 + 18 + 3

= 309