First slide

Permutations

Question

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

Moderate

A

NAAGI
B

NAAIG
C

NIAAG
D

NAIAG

Solution

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

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