On Combinatorics and Permutations I am working on the following question and would like to know if I am correct/ on the right track. 
1) How many possible ways are there to form five-letter words using only the letters A–H? 
2) How many such words consist of five distinct letters?
My work:
1) Between A to H there are 8 letters and I have 5 spots. I can repeat letters, thus I have $8\times8\times8\times8\times8 $ this I will then divide by the 5! which is the ways they can be arranged. Final answer $8^5/120$.
2) Now that the letters have to be different I will only have $8\times7\times6\times5\times4$ divided by 120 which is the amount of ways variables can be repeated. 
 A: The division by $5!$ is entirely unnecessary.  This should be obviously wrong by the fact that your proposed answer is not even an integer and as such could not have possibly been the answer to a counting question.
Your concern about the difference between words $A_1A_2A_3A_4A_5$ and $A_1A_2A_3A_5A_4$ is unfounded.  In the process of applying rule of product to count the number of options for our word we applied the following steps:


*

*Pick the letter that appears in the first position of the word

*Pick the letter that appears in the second position of the word

*$\vdots$

*Pick the letter that appears in the last position of the word
Here, there is only one sequence of choices that results in the word $AAAAA$, namely the sequence of choices where we said the letter $A$ in every step.  We did not over count the word $AAAAA$ multiple times, we counted it exactly once.
Now, you might have decided to "pick a letter, then pick where it goes"  followed by "pick another letter and then pick where it goes" which would have given $(8\times 5)\times (8\times 4)\times (8\times 3)\times (8\times 2)\times (8\times 1)$, in which case we would have wanted to divide by $5!$, but we didn't do that.  We very explicitly in our first step decided what the first letter was going to be.
