Question

The letters of the word MODESTY are written in all possible orders and these words are written out as in a dictionary then the rank of the word MODESTY is

• A. 5040
• B. 720
• C. 1681
• D. 2520

Hint:

The words in a dictionary are arranged in an alphabetical manner.

Answer 1

The Correct Option is C

The words in a dictionary are arranged in an alphabetical manner.

Words starting with D are 6! = 720, start with E are 720. start with MD are 5! = 120 and start with ME are 120. Now the first word that starts with MO is nothing but MODESTY. Hence rank of MODESTY is 1681.

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Alternate Approach-1

The number of words starting with D, arranging the other 6 letters = 6! =720

The number of words starting with E = 6! = 720

The number of words starting with M = 6! = 720 but one of these words is MODESTY

The number of words starting with MD = 5! =120

The number of words starting with ME = 5! =120

Now the first-word start with MO is MODESTY. 

Hence, the rank of MODESTY = 720 + 720 + 120 + 120 + 1 =1681

Read More: Permutations and Combinations

Answer 2

In a dictionary, the words are arranged in alphabetical order at every stage.
Starting with the letter D and arranging the remaining 6 letters, we get 6! = 6×5×4×3×2×1 = 720.
Then, starting with the letter E and arranging the other six letters D, M, O, S, T, Y in different ways, we get 6! = 720
The number of words starting with M are 6! = 720 but one of these words is the word ‘MODESTY’ itself.
Therefore, we first determine the number of words starting with MD, ME, MO
Count of words starting with MD = 5! = 120
Count of words starting with ME = 5! = 120
Now, the initial word that starts with MO is noting but MODESTY.
Therefore, rank of MODESTY is :
= 720 + 720 + 120 + 120 + 1 = 1681.
So, the correct option is (C) : 1681.