Question

If all the letters of the word $AGAIN$ be arranged as in a dictionary, what is the fiftieth word?

Answer 1

In dictionary the words at each stage are arranged in alphabetical order. Starting with the letter $A,$ an 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 $\frac{4!}{2!}=\frac{4\times 3\times 2}{2}=12$ 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 $\frac{4!}{2!}=\frac{4\times 3\times 2}{2}=12$ ways. So, there are $12$ words, which start with $I$.

Thus, we have so far constructed $48$ words.

The ${49}^{th}$ word is $NAAGI$ and hence the ${50}^{th}$ word is $NAAIG$.