Question

If the words formed by using the letters of the word ‘AGAIN’ are arranged in the form of dictionary then $50^{th}$ word is
A. NAAGI
B. NAAIG
C. NAGAI
D. NAIAG

Answer 1

We will use permutation to solve the type of question. So we will go alphabetically for the word, as in the same way written in the dictionary. Then using the permutation we will go to the number close to 50th word or on it whichever is possible and count till we get 50th word.

Complete step by step answer:
Moving ahead with the question, we have the word ‘AGAIN’ so let us first write it in alphabetical order i.e.
A, A, G, I, N
So first let us choose the first alphabet i.e. A and using the permutation let us find out the total number of words that will be ,made, so as we have four words other than first alphabet A, so according to permutation possible number of words will be$4!$. i.e.
Words with first alphabet ‘A’ will be$=4!=4\times 3\times 2\times 1=24$
So there will be 24 words with the first letter ‘A’. But we have to find 50th word so we should now move to the second letter. Which is ‘G’. So total number of words that will form when first alphabet is ‘G’ $G,\underline{{}},\underline{{}},\underline{{}},\underline{{}}$ so there will be again 4 letters that will help in making word. But there is one word repeated i.e. ’A’ makes two words the same, so in order to find different words we have to divide it by 2. So total number of words we will get when first alphabet is ‘G’ is;
Words with first alphabet ‘G’ will be$=\dfrac{4!}{2}=\dfrac{4\times 3\times 2\times 1}{2}=12$
So the total word we will get from ‘A’ and ‘G’ is$24+12=36$, which is still less than 50th word. So we should move further with next word i.e. ‘I’ it again have same case as we have for words starting with ‘G’ i.e. it had four words which will come after ‘I’ from which two alphabet are same i.e. ‘A’ so again going with same condition we will have;
Words with first alphabet ‘I’ will be$=\dfrac{4!}{2}=\dfrac{4\times 3\times 2\times 1}{2}=12$
So now the total words from ‘A’, ‘G’ and ‘I’ is$24+12+12=48$ which is very close to 50th word. So we need two more words. Now let us count them simply. So the next word will start, whose first alphabet is ‘N’. So 1st word with ‘N’ will be; NAAGI, as arranged in alphabet order. Which is 49th word.
And the next 50th word will be ‘NAAIG’. As arranged in alphabet order which will come after the word ‘NAAGI’
So we can say that ‘NAAIG’ is the 50th word when arranged in the form of a dictionary.
So, the correct answer is “Option B”.

Note: Never forget to divide the possible permutations you got when they have some repetition. As in our case the alphabet ‘A’ is repeated two times so we divide the result by two, if it gets repeated by 3 three times then divide it by 3 and so on.