Question

The letters of the word “PATNA” are arranged in all possible ways as in a dictionary, then the rank of the word “PATNA” is:

 $
  A.41 \\
  B.42 \\
  C.43 \\
  D.44 \\
 $

Answer 1

Hint: Combination is the number of ways in which definite objects/letters can be arranged in a defined manner. In this question, the letters of the word “PATNA” to be arranged in the alphabetical order and rank of the word is needed to be determined for which we need to use the concept of combination by evaluating the number of words formed by keeping every letter at consequent positions.

Complete step-by-step answer:
The letters of the word “PATNA” are arranged in alphabetical order as A A P T N

The number of words starting with the letter “A” can be calculated $ 4! = 24 $ .
Similarly,
The number of words starting with the letter “N” can be calculated as $ \dfrac{{4!}}{{2!}} = \dfrac{{24}}{2} = 12 $ . Here, 2! I divided as the letter “A” is repeated 2 times in the word “PATNA”.
Again,
The number of words starting with the letter “PAA” can be calculated as $ 2! = 2 $ .
Again,
The number of words starting with the letter “PAN” can be calculated as $ 2! = 2 $ .
Again,
The number of words starting with the letter “PATA” can be calculated as $ 1! = 1 $ .

Till the starting of the words with the letter “T”, a sum of $ 120 + 120 + 120 + 120 + 120 = 600 $ words has been made.

Till now, the total number of words formed is $ 24 + 12 + 2 + 2 + 1 = 41 $ , but we need to determine the position of the word “PATNA” for which we need to add 1 to 41 as: $ 41 + 1 = 42 $ .
So, the correct answer is “Option B”.

Note: Candidates should be aware while using the ascending order of the alphabets and while using the formula of combinations. Before actually assigning the order number to the required word one should not miss on any of the previous combinations of the words.