Question

If the letters of the word ‘NAAGI’ are arranged as in a dictionary then the rank of the given word is:
(a) 23
(b) 84
(c) 49
(d) 48

Answer 1

Hint: Find out the number of words starting with A, I and G and then the next word will be ‘NAAGI’. So after finding out the number of words before ‘NAAGI’ using permutation and combinations, add 1 to the result to get the rank of ‘NAAGI’.

Complete step-by-step answer:
Rank of a word means that we are supposed to find the position of that word when all permutations of the word are written in alphabetical order. Follow the steps to find the rank of any word:

Step I: Write down the letters in alphabetical order.

Step II: Find the number of words that start with a superior letter
.
Step III: At last add 1 to the total number of words obtained to get the rank of the required word.

Now, let us come to the question. We have to find the position of ‘NAAGI’.

Number of words that start with A $=4!$

Number of words that start with G $=\dfrac{4!}{2!}$

Number of words that start with I $=\dfrac{4!}{2!}$

Now, the next word will be ‘NAAGI’ according to the alphabetical order. Therefore, the rank of this word:

$\begin{align}
  & =4!+\dfrac{4!}{2!}+\dfrac{4!}{2!}+1 \\
 & =24+12+12+1 \\
 & =49 \\
\end{align}$

Hence, the correct answer is option (c).

Note: When we are finding the number of words that are starting with G and I, we have divided the number by 2!, that’s because A is repeating twice. Also, 1 is added in the last because all the steps above are to find the number of words that will appear before ‘NAAGI’, so ‘NAAGI’ will appear after the 48th word, that is, at 49th place.