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: We start solving the problem by recalling the order of the first letters in the dictionary. We then find the total number of words formed with the first letter ‘A’ using the fact that the number of ways of arranging ‘n’ places with ‘n’ different objects is $n!$ ways. We then find the total number of words formed with the first letter ‘A’ using the fact that the number of ways of arranging ‘n’ places with ‘r’ repeating objects and remaining is non-repeated is $\dfrac{n!}{r!}$ ways. We then add 1 to the number of words obtained to get the rank of the word ‘NAAGI’ as we can see that this is the first word formed with the first letter ‘N’.

Complete step by step answer:
According to the problem, we are asked to find the rank of the word ‘NAAGI’ if the letters of this word are arranged as in a dictionary.
We know that the first letters of the words in the dictionary are in the order A, G, I, N.
So, let us find the words formed with ‘A’ as the first letter in the 5 letter words.

A

We need to arrange the remaining 4 places with 4 different letters. We know that the number of ways of arranging ‘n’ places with ‘n’ different objects is $n!$ ways.
So, we get the total number of words with the first letter ‘A’ as $4!=24$.
So, let us find the words formed with ‘G’ as the first letter in the 5 letter words.

G

We need to arrange the remaining 4 places with 2 non-repeating letters and one repeating letterer. We know that the number of ways of arranging ‘n’ places with ‘r’ repeating objects and remaining non-repeated is $\dfrac{n!}{r!}$ ways.
So, we get the total number of words with the first letter ‘G’ as $\dfrac{4!}{2!}=\dfrac{24}{2}=12$.
So, let us find the words formed with ‘I’ as the first letter in the 5 letter words.

I

We need to arrange the remaining 4 places with 2 non-repeating letters and one repeating letterer. We know that the number of ways of arranging ‘n’ places with ‘r’ repeating objects and remaining non-repeated is $\dfrac{n!}{r!}$ ways.
So, we get the total number of words with the first letter ‘I’ as $\dfrac{4!}{2!}=\dfrac{24}{2}=12$.
So, the words formed with the first letters ‘A’, ‘G’, and ‘I’ are $24+12+12=48$.
Now, we can see that the letters in NAAGI are arranged in dictionary order other than N. So, this will be the first word of those which were formed with the first letter as ‘N’.
So, we can find the rank of the word ‘NAAGI’ by adding 1 to the 48 i.e., 49.
$\therefore $ The rank of the word ‘NAAGI’ is 49.

$\therefore $ The correct option for the given problem is (c).

Note:
Here we have considered only 5 letter words that were formed with the letters given otherwise the answer would be different. We can also solve this problem as shown below:
Let us find the 5 letter words that can be formed with the given letters i.e., $\dfrac{5!}{2!}=\dfrac{120}{2}=60$.
Now, let us find the 5 letter words that can be formed with the first letter ‘N’.

I

So, we get the total number of words with the first letter ‘I’ as $\dfrac{4!}{2!}=\dfrac{24}{2}=12$.
Now, the number of words that were formed with the first letter as ‘A’, ‘G’, and ‘I’ will be $60-12=48$ words.
Since ‘NAAGI’ is the first word formed with the first letter ‘N’, we get the rank of ‘NAAGI’ as $48+1=49$.