If all the letters of the word 'AGAIN' be arranged as in a dictionary, then the.fiftieth word is 

In dictionary the words at each stage are arranged in alphabetical order. Starting with the letter A, and arranging the other four letters GAIN, we obtain 4! = 24 words. Thus, there are 24 words which start with A. These are the first 24 words. Then, starting with G, and arranging the other four letters A. A, I, N in different ways, we obtain $\frac{4!}{2!}=\frac{24}{2}=12$ words.

Thus, there are 12 words, which start with G. Now, we start with I. The remaining 4 letters A, G, A, N can be arranged in $\frac{4!}{2!}=12$ ways.

 So, there are 12 words, which start with I the remaining 4 letters.Thus we have so far constructed 48 words. The 49th word is NAAGI and hence the 50h word is NAAIG