Determine the total number of $4$-digit numbers which can be obtained using the digits $1, 2, 3, 4, 5$. It's a basic question, but I don't know why I am getting confused.
Determine the total number of $4$-digit numbers which can be obtained using the digits $1, 2, 3, 4, 5$. Also find how many of them are divisible by $4$.
 A: Case 1: digits cannot be repeated
For the first question, there are five choices for each of the four digits.  
For the second question, there are five choices for the thousands digit and five choices for the hundreds digit.  For a number to be a multiple of $4$, the last two digits must be a multiple of $4$.  How many such multiples of $4$ are there between $11$ and $55$ inclusive?
Case 2:  digits cannot be repeated
For the first question, there are five choices for the thousands digit, four choices for the hundreds digit, three choices for the tens digit, and two choices for the units digit.
For the second question, how many multiples of $4$ that lie between $11$ and $55$ inclusive do not contain a repeated digit?  Since each of these choices require the use of two distinct digits, there are three choices for the thousands digit and two choices for the hundreds digit.
A: How many ways can you pick a digit to not use?
For each of these, how many ways can you arrange the remaining four digits?
A number is divisible by $4$ if and only if the last two digits (on the right) are divisible by 4.
There are twenty possible two-digit pairs, from $12$ to $54$.  How many of them are divisible by $4$?
