VerifiedHint: Combination is the number of ways in which definite objects/letters can be arranged in a defined manner. Here, in this question, the rank of the word “NAVEEN” have to be determined such that all words that can be formed using the letters are ranged in alphabetical order for which we need to start evaluating the number of the words coming before the word “NAVEEN” in the ascending order of the alphabets.
Complete step by step solution:
The letters of the word “NAVEEN” are arranged in alphabetical order as “A E E N N V”.
Now, according to the question, the alphabets are arranged in ascending order and follow the concept of a dictionary. So, here, after the letter “A”, two letters are repeating themselves i.e., “E” and “N”.
The number of words starting from the letter “A” is evaluated by the concept of combination as:
$\dfrac{{5!}}{{2! \times 2!}} = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} = 30 - - - - (i)$
Here, 5! is used for the total number of letters after A and 2!s are used for the repeating letters.
Similarly, after the letter “A”, letter “E” will occupy the first position in the word and the number of the words starting from the letter “E” is evaluated by the concept of combination as:
$\dfrac{{5!}}{{2!}} = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{ 2 \times 1}} = 60 - - - - (ii)$
Again here, 5! is used for the total number of letters after E and 2! is used for the repeating letters. Also notice here that no E will repeat its word.
Again, after the letter “E”, letter “N” will occupy the first position in the word following the letters “A” and so, the number of the words starting from the letter “NA” is evaluated by the concept of combination as:
$\dfrac{{4!}}{{2!}} = \dfrac{{ 4 \times 3 \times 2 \times 1}}{{ 2 \times 1}} = 12 - - - - (iii)$
Again here, 4! is used for the total number of letters after NA and 2! is used for the repeating letters of E. Also notice here that no N will repeat its word.
Again, after the letter “NA”, letter “N” will occupy the first position in the word following the letters “A” and “E” and so, the number of the words starting from the letter “NAE” is evaluated by the concept of combination as: $3! = 6 - - - - (iv)$
Here, the spelling of the word starting with “N” resembles the original word “NAVEEN”.
Again, after “NAE___”, “NAN___” will be followed and so the number of the words starting from the letter “NAN” is evaluated by the concept of combination as:
$\dfrac{{3!}}{{2!}} = \dfrac{{3 \times 2 \times 1}}{{2 \times 1}} = 3 - - - - (v)$
here, 3! is due to the number of the succeeding letters and 2! Is due to the repeating letters.
At last, the original word will be followed by the “NAV___” i.e., “NAVEEN” for which the count is 1 - - - (vi)
Now, adding all the equations (i), (ii), (iii), (iv), (v) and (vi) to determine the rank of the word “NAVEEN” in the dictionary as:
Rank = 30 + 60 +12 + 6 + 3 + 1 = 112
Hence, the rank of the word “NAVEEN” when all words that can be formed using the letters are ranged in alphabetical order is 112.
Note: Candidates should be aware while using the ascending order of the alphabets and while using the formula of combinations.
