Question

If different words are formed with all the letters of the word ‘AGAIN’ and are arranged alphabetically among themselves as in a dictionary, the word at $50^{th}$ place will be:
A. NAAGI
B. NAAIG
C. IAAGN
D. IAANG

Answer 1

To solve this question, we will make use of the concept of permutations and combinations. We will see how many words are formed from each letter alphabetically by fixing the first position of the word by that letter and then the number of ways in which the next positions can be filled will be equal to the number of words starting from that letter. Then we will see under which word the 50th word falls. Then we will arrange the words from that letter in alphabetical order and the word which will come on the $50^{th}$ position will be the answer.

Complete step-by-step solution
Here, we will see how many words can be formed from each letter according to the dictionary.
Now, the word given to us is ‘AGAIN’. We know that the first letter when arranged in alphabetical order in this word is ‘A’.
Thus, the number of words which can be formed from letter ‘A’ are given as follows:
We have fixed the first position among the 5 positions in this word by letter ‘A’ as follows:
$\underline{A}\text{ }\underline{\text{ }}\text{ }\underline{\text{ }}\text{ }\underline{\text{ }}\text{ }\underline{\text{ }}$
Now, in the next 4 positions, there are all distinct letters and we know that n distinct objects can be arranged in n! ways. Thus, the number of words which can be formed from letter ‘A’ are:
$\begin{aligned} & 4! \\\ & \Rightarrow 24 \\\ \end{aligned}$
Now, the second letter in this word when arranged alphabetically is ‘G’.
Thus, the number of word which start from letter ‘G’ are given as follows:
We have fixed the first position among the 5 positions in this word by letter ‘G’ as follows:
\underline{G}\text{ }\\!\\!\\_\\!\\!\text{ }\\!\\!\\_\\!\\!\text{ }\\!\\!\\_\\!\\!\text{ }\ \text{ }\\!\\!\\_\\!\\!\text{ }
Now, in the next 4 positions, there are 2 similar letters (2 As) and we know that the number of ways to arrange ‘n’ objects in which ‘r’ are different is $\dfrac{n!}{r!}$. Thus the number of words which can be formed from the letter ‘G’ are:
$\begin{aligned} & \dfrac{4!}{2!} \\\ & \Rightarrow \dfrac{24}{2} \\\ & \Rightarrow 12 \\\ \end{aligned}$
Now, we can see that the total words in the dictionary now are $24+12=36$.
Now, the third letter in this word when arranged alphabetically is ‘I’.
Thus, the number of words which start from letter ‘I’ are given as follows:
We have fixed the first position among the 5 positions in this word by letter ‘I’ as follows:
\underline{I}\text{ }\\!\\!\\_\\!\\!\text{ }\ \text{ }\\!\\!\\_\\!\\!\text{ }\ \\_\ \\_
Now, in the next 4 positions too there are 2 similar letters (2 As). Thus, the number of words which start from the letter ‘I’ are:
$\begin{aligned} & \dfrac{4!}{2!} \\\ & \Rightarrow \dfrac{24}{2} \\\ & \Rightarrow 12 \\\ \end{aligned}$
Now, we can see that the total letters in the dictionary are $36+12=48$.
Hence, the 50th letter will now start from ‘N’ and be the second word in alphabetical order.
In the first word, the first position will be fixed by ‘N’ as follows:
\underline{N}\ \text{ }\\!\\!\\_\\!\\!\text{ }\ \text{ }\\!\\!\\_\\!\\!\text{ }\ \text{ }\\!\\!\\_\\!\\!\text{ }\ \text{ }\\!\\!\\_\\!\\!\text{ }
Now, in the remaining letters, alphabetically, the first letter is ‘A’ which comes twice. Thus, the 49th word will thus become:
\underline{N}\ \underline{A}\ \underline{A}\ \\_\ \\_
Now, the remaining letters are ‘G’ and ‘I’ in which ‘G’ occurs first. Thus, the 49th word is:
$\underline{N}\ \underline{A}\ \underline{A}\ \underline{G}\ \underline{I}$
Now, for the 50th letter, only the last alphabetical order will be reversed. Hence, the 50th word is given as:
$\underline{N}\ \underline{A}\ \underline{A}\ \underline{I}\ \underline{G}$
Thus, the 50th word is NAAIG.
Hence, option (B) is the correct option.

Note: These types of questions are very important and to be very tricky. Here, we have been asked the word on the 50th position when words formed from the letters of this word are arranged in a dictionary, but the question can also ask us on which position the given word itself will lie. These two are the most common dictionary type questions which we should know properly.