Question

Find the number of words 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 fiftieth word?

Answer 1

If we were to have a $5$-letter word with each letter being different, we could make:

$5!=120$ words.

However, since there are $2$ As, we need to divide by the internal ordering the As can do ($2!$) to eliminate duplicates. So in total we have:

$\frac{5!}{2!}=\frac{120}{2}=60$words.

Will the $50th$ word start with N? To see, let's pin the N as the first letter and then see the number of words we can make from the remaining letters:

$\frac{4!}{2!}=\frac{24}{2}=12$ and so yes, the $50th$ word will start with $N$ (N starts the words $49-60$).

and, the answer is NAAIG