Question

Find the number of words, with or without meaning, which can be made using all letters of word AGAIN. If these words are written as in a dictionary, what will be the 50th word?

Answer 1

NAAIG

Explanation:

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:

`(5!)/(2!)=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:

`(4!)/(2!)=24/2=12` and so yes, the 50th word will start with N (N starts the words 49 - 60).

The 12 words starting with N are:

NAAGI
`color(red)("NAAIG")`
NAGAI
NAGIA
NAIAG
NAIGA
NGAAI
NGAIA
NGIAA
NIAAG
NIAGA
NIGAA