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'.

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Hint: In the word 'MOTHER' we can see that there are a total of six letters, so, first of all we will write all the letters of the word in alphabetical order and then we will start making words according to the rule of the dictionary.

Complete step-by-step answer:
We have been asked to find the rank of the word 'MOTHER' when the letter of the words are written in all possible orders and these words are written out as in a dictionary.
Letters in the word 'MOTHER' are in the order E, H, M, O, R, T according to the rules of the dictionary.
Now, we have different cases as follows:
Case I- Words with E _ _ _ _ _
5 letters can be arranged as 5! at 5 places.
\[\Rightarrow \text{Words with E }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 5!}\]
Case II- Words with H _ _ _ _ _
\[\Rightarrow \text{Words with H }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 5!}\]
Case III- Words with ME _ _ _ _
4 letters can be arranged by 4! at 4 different places.
\[\Rightarrow \text{Words with ME }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 4!}\]
Case IV- Words with MH _ _ _ _
\[\Rightarrow \text{Words with MH }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 4!}\]
Case V- Words with MOE _ _ _
We can arrange 3 letters by 3! at 3 different places.
\[\Rightarrow \text{Words with MOE }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 3!}\]
Case VI- Words with MOH _ _ _
\[\Rightarrow \text{Words with MOH }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 3!}\]
Case VII- Words with MOR _ _ _
\[\Rightarrow \text{Words with MOR }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 3!}\]
Case VIII- Words with MOTE _ _
We can arrange 2 letters by 2! at 2 places.
\[\Rightarrow \text{Words with MOTE }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ = 2!}\]
After this, we get the word 'MOTHER'.
On adding all words in different cases, we get:
  & \Rightarrow 5!+5!+4!+4!+3!+3!+3!+2! \\
 & \Rightarrow 120+120+24+24+6+6+6+2 \\
 & \Rightarrow 308 \\
So, there are 308 words before the word 'MOTHER'.
Therefore, \[{{309}^{th}}\] word will be 'MOTHER' which is the rank of the word in the dictionary.

Note: Remember that, in the question, related to the rank of a word in a dictionary, first of all we must have to arrange the letter of the word in alphabetically order and then take the different cases until you get the desired word.
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