Question

If the letter of the word “MOTHER” is written in all possible orders and these words are written out as in a dictionary, Then the rank of the word MOTHER is?
A) 307
B) 261
C) 308
D) 309

Answer 1

To find the rank of the word Mother we will have to evaluate the no of ways of each word in the dictionary in alphabetic order-wise by using the concept of permutation. We will start evaluating the number of words starting with E and go further until we do not get the required rank of the word MOTHER. We take the formula: $n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times \left( {n - 3} \right) \times ........ \times 1$

Complete step by step answer:
There are six letters present in the word MOTHER.
In dictionary order letters should be arranged in alphabetical order.
So, the pattern will be E, H, M, O, R and T.
Number of words starts with E $= 5! = 120$
Number of words starts with H $= 5! = 120$
Number of words starts with ME $= 4! = 24$,
Now, Number of words starts with MH $= 4! = 24$
Number of words starts with MOE $= 3! = 6$
Number of words starts with MOH $= 3! = 6$
Number of words starts with MOR $= 3! = 6$,
Number of words starts with MOTE $= 2! = 2$,
Number of words start with MOTHER $= 1! = 1$,

The first word to start with MOTH is MOTHER.
Now in order to calculate the rank of mother, we have to add all the possible number of ways up to the words MOTHER.
So rank of the word MOTHER $= 5! + 5! + 4! + 4! + 3! + 3! + 3! + 2! + 1!$
$\Rightarrow 2 \times 5! + 2 \times 4! + 3 \times 3! + 2! + 1! = 2 \times 120 + 2 \times 24 + 3 \times 6 + 2 + 1$
$\Rightarrow 2 \times 120 + 2 \times 24 + 3 \times 6 + 2 + 1 = 309$
Hence, the required rank of the word MOTHER is $309.$

Therefore, Option (D) is the correct answer. The rank of the word MOTHER is 309.

Note:
We should be careful to understand the concept of arrangement before solving the problem. In permutation we apply the concept of arrangement while in combination we simply apply the concept of selection.