Question

If the letters of the word MOTHER are arranged in all possible orders, and these words are written as in a dictionary, then the rank of the word MOTHER will be
(a) 240
(b) 261
(c) 308
(d) 309

Answer 1

Here, you need to find the rank of the word MOTHER. In a dictionary, words are arranged in alphabetical order. We will write the letters of the word MOTHER in alphabetical order. We will find the words that will come before the word MOTHER. We will then find the number of these words and add them. The rank of the word MOTHER will be one more than the number of words that come before it.
Formula Used: The number of ways in which $n$ objects can be arranged without repetition is given by the formula $n!$.

Complete step by step solution:
We know that in a dictionary, words are arranged in alphabetical order.
First, we will arrange the letters of the word MOTHER in alphabetical order.
Thus, we get E, H, O, M, R, T.
We will use the formula for the number of ways without repetition to find the rank of the word MOTHER in the dictionary.
The number of ways in which $n$ objects can be arranged without repetition is given by the formula $n!$.
Now, we know that the words starting with E will appear first in the dictionary.
Let us fix letter E as the first letter of the word.
The number of ways in which the other 5 letters can be arranged is given by $5! = 120$ ways.
Therefore, there are 120 words that start with E, and contain the letters of the word MOTHER.
Next, the words starting with letter H will appear in the dictionary.
Let us fix letter H as the first letter of the word.
The number of ways in which the other 5 letters can be arranged is given by $5! = 120$ ways.
Therefore, there are 120 words that start with H, and contain the letters of the word MOTHER.
Next, the words starting with the letter M will appear in the dictionary.
We will fix letter M as the first letter of the word.
Since the word MOTHER also starts with M, we do not need to arrange the remaining 5 letters in every possible order.
We know that the words starting with ME will come before the words starting with MH, MO, MR, and MT.
Thus, we will fix the second letter of the word as E, H, O, R, and T one by one.
Let us fix the letter M as first letter and letter E as second letter.
The number of ways in which the other 4 letters can be arranged is given by $4! = 24$ ways.
Therefore, there are 24 words that start with ME, and contain the letters of the word MOTHER.
Next, let us fix the letter M as first letter and letter H as second letter.
The number of ways in which the other 4 letters can be arranged is given by $4! = 24$ ways.
Therefore, there are 24 words that start with MH, and contain the letters of the word MOTHER.
Now, we will fix the letter M as first letter and letter O as second letter.
Since the word MOTHER also starts with MO, we do not need to arrange the remaining 4 letters in every possible order.
We know that the words starting with MOE will come before the words starting with MOH, MOR, and MOT.
Thus, we will fix the third letter of the word as E, H, R, and T one by one.
Let us fix the letter M as first letter, letter O as second letter, and letter E as third letter.
The number of ways in which the other 3 letters can be arranged is given by $3! = 6$ ways.
Therefore, there are 6 words that start with MOE, and contain the letters of the word MOTHER.
Now, let us fix the letter M as first letter, letter O as second letter, and letter H as third letter.
The number of ways in which the other 3 letters can be arranged is given by $3! = 6$ ways.
Therefore, there are 6 words that start with MOH, and contain the letters of the word MOTHER.
Next, let us fix the letter M as first letter, letter O as second letter, and letter R as third letter.
The number of ways in which the other 3 letters can be arranged is given by $3! = 6$ ways.
Therefore, there are 6 words that start with MOR, and contain the letters of the word MOTHER.
Now, we will fix the letter M as first letter, letter O as second letter, and letter T as third letter.
Since the word MOTHER also starts with MOT, we do not need to arrange the remaining 3 letters in every possible order.
We know that the words starting with MOTE will come before the words starting with MOTH and MOTR.
Thus, we will fix the fourth letter of the word as E, H or T one by one.
Let us fix the letter M as first letter, letter O as second letter, letter T as third letter, and letter E as fourth letter.
The number of ways in which the other 2 letters can be arranged is given by $2! = 2$ ways.
Therefore, there are 2 words that start with MOTE, and contain the letters of the word MOTHER.
Now, we will fix the letter M as first letter, letter O as second letter, letter T as third letter, and letter H as fourth letter.
Since the word MOTHER also starts with MOTH, we do not need to arrange the remaining 2 letters in every possible order.
We know that the word starting with MOTHE will come before the word starting with MOTHR.
The only word that starts with MOTHE and contains the letters of the word MOTHER, is the word MOTHER itself.
Finally, we will add the number of all the words that come before the word MOTHER in the dictionary.
Number of words that come before the word MOTHER $=$ Number of words starting with E $+$ Number of words starting with H $+$ Number of words starting with ME $+$ Number of words starting with MH $+$ Number of words starting with MOE $+$ Number of words starting with MOH $+$ Number of words starting with MOR $+$ Number of words starting with MOTE
Therefore, we get
Number of words that come before the word MOTHER $= 120 + 120 + 24 + 24 + 6 + 6 + 6 + 2 = 308$
Since 308 words come before the word MOTHER, the word MOTHER is the ${309}^{\text{th}}$ word.
Therefore, the rank of the word MOTHER is 309 when the letters of the word MOTHER are arranged in all possible orders and written as in a dictionary.

Thus, the correct option is option (d).

Note:
We should remember that the question asks for the rank of the word MOTHER, and not the number of words that come before it. A common mistake is to leave the answer at 308 and mark option (c) as the correct answer. That is incorrect.