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This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. 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 *ATTEMPT**
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 Answer Hint: The words in a dictionary are arranged in alphabetical order at each stage. In this question must consider all the words in order starting with A, G, I, N. As in dictionary A will appear at first place so remaining four letters can be filled in how many ways. The number of ways of arranging the remaining 4 letters taking all at a time i.e. the number of words starts with A are $4!$. Similarly, you can find the number of words formed with other letters at first place. We have to find the word at 50th place in the dictionary according to given alphabets. Complete step-by-step answer: As the are placed in alphabetical order so the number of words start with letter A is given by:\[ \Rightarrow \]Number of words that can be formed with A at first position $ = 4! = 24$.Rest of the letters can be placed at first place and let us see how many words are formed from them respectively. G letter will come after A letter. So, number of words formed with first letter G is given by\[ \Rightarrow \]Number of words that can be formed with G at first position $ = \dfrac{{4!}}{{2!}} = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 12$.Here you can see we divide $4!$ by $2!$ why? The reason behind that is that there are two A letters in the word given to us. So, to avoid repetition of words and the position of the last word appear in a dictionary whose first letter G can be given by the addition of number of words formed by A and I i.e.$ \Rightarrow $The number of words till G letter at starting = $24 + 12 = 36$.Similarly, the number of words formed with first letter I is given by \[ \Rightarrow \]Number of words that can be formed with I at first position $ = \dfrac{{4!}}{{2!}} = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 12$.Hence, you can find the position of last word start with letter I i.e.$ \Rightarrow $ The number of words till I letter at starting = $24 + 12 + 12 = 48$.Now you can see the first 48 words appearing in a dictionary are from the words starting with letter A, I, G. After that the next two words i.e. word at 49th and 50th position have starting letter N.So, look at the first two words that appear in the dictionary starting with the N alphabet are NAAGI and NAAIG in alphabetically order respectively.Here, the word at 50th position is NAAIG.Hence the correct option is C. Note: The common mistake done by students in calculating the number of ways of arranging the letters is repetition when we arrange a similar alphabet the word form is the same. So, we have to divide those repeated words. If a similar alphabet repeats then the number of ways of arranging them is divided with the total ways. Here in this question, the number of letters with A is 2 so divide it by $2!$. Open in App
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If all the letters of the word again be arranged as per english dictionary, what is the 50th word
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