What will be the 50th word if the permutations of the word again are written in a dictionary form

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I have to Find the 50th word of all permutations of word AGAIN ,when arranged accord to dictionary.?There are 60 words with or without meaning ,but how do i find out 50th word .Thanks

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Last updated at Jan. 30, 2020 by Teachoo

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Example 22 Find the number of words with or without meaning which can be made using all the letters of the word AGAIN. If these words are written as in a dictionary, what will be the 50th word? ‘AGAIN’ = 2A, 1G, 1I & 1N In dictionary, letters appear alphabetically, 4 letters in which A, G, I, N Since, we arrange 4 letters, Number of words = 4P4 = 4! = 24 4 letters in which 2A, I, N Since, letters are repeating, number of words = 𝑛!/𝑝1!𝑝2!𝑝3! Number of letter = n = 4 Since 2A, p1 = 4 Number of words = 4!/2! = 12 4 letters in which 2A, G, N Since, letters are repeating, number of words = 𝑛!/𝑝1!𝑝2!𝑝3! Number of letter = n = 4 Since 2A, p1 = 4 Number of words = 4!/2! = 12 Thus, Total no of words starting with A, G, & I = 24 + 12 + 12 = 48 Hence, 49th word will be start from N i.e. N A A G I & remaining four rearrange according to a dictionary Thus, The 50th word is N A A I G Hence the 50th word will be NAAIG

Text Solution

NAAGINAAIGNIAAGNAIAG

Answer : B

Solution : In dictionary the words at each stage are arranging the other four letters GAIN, we obtain 4!=24 words. <br> Thus, there 24 words which start with A. These are the first 24 words. <br> Then, starting with G, and arranging the other four letters A, A, I, N in different ways, we obtain `(4!)/(2!)=(24)/(2)=12` words. <br> Thus, there are 12 words, which start with G. <br> Now, we start with I. the remaining 4 letters A, G, A, N can be arranged in `(4!)/(4!)=12` ways. So, there are 12 words, which start with I, Thus, we have so far constructed 48 words. <br> The 49th word is NAAGI and hence the 50th word is NAAIG.