SOLUTION: Find the number of 4 digt numbers that can be made with 1,2,3,4,& 5 in which at least two digits are identical.

First we'll calculate the number of ALL 4-digit numbers that can
be made with them.  Then we'll calculate and subtract the number
of 4-digit numbers that don't have any of the digits identical.

To find all possible 4-digit numbers that can be made.

1. There are 5 ways to choose the 1st digit.
2. For each of those 5 ways to choose the 1st digit, there are 5 ways to choose the 2nd digit. 
   That's 5*5 or 52 or 25 ways to choose the 1st 2 digits.
3. For each of those 5*5 or 52 or 25 ways to choose the 1st 2 digits, there are 5 ways 
   to choose the 3rd digit. That's 5*5*5 or 53 or 125 ways to choose the 1st 3 digits.
4. For each of those 5*5*5 or 53 or 125 ways to choose the 1st 3 digits, there are 5 ways 
   to choose the 4th digit. That's 5*5*5*5 or 54 or 625 ways to choose the 1st 4 digits,
   which is all of them.


Now we must subtract the number of ways in which there are no two digits alike.
So we'll calculate that number of 4-digit numbers that we must subtract:

1. There are 5 ways to choose the 1st digit.
2. For each of those 5 ways to choose the 1st digit, there are 4 ways to choose the 2nd digit. 
   That's 5*4 or 20 ways to choose the 1st 2 digits.
3. For each of those 5*4 or 20 ways to choose the 1st 2 digits, there are 3 ways 
   to choose the 3rd digit. That's 5*4*3 or 60 ways to choose the 1st 3 digits.
4. For each of those 5*4*3 or 60 ways to choose the 1st 3 digits, there are 2 ways 
   to choose the 4th digit. That's 5*4*3*2 or 120 ways to choose the 1st 4 digits,
   which is all of them.

So the answer is 54 - 5*4*3*2 or 625 - 120 or 505.

Edwin