Whether you really want a list of all the different ways, or whether you only want to know HOW MANY ways there are, I will show you HOW to solve the problem and leave the details to you.
(1) Since the $10 bills and $5 bills together make a total number of dollars that is a multiple of 5, and since the total $100 is a multiple of 5, the total value of the $1 bills must be a multiple of 5.
(2) For each number of $1 bills that is a multiple of 5, the number of ways to make change for $100 is determined by the number of $10 bills you can have. For example, if there are 55 $1 bills, then the remaining amount is $45, and you can use 0, 1, 2, 3, or 4 $10 bills, making up the rest with $5 bills. That makes 5 ways to make change for $100 if the number of $1 bills is 55.
To outline the complete solution, then....
column 1: # of $1 bills
column 2: remaining amount
column 3: # of choices for the number of $10 bills
0 100 11 (0 to a maximum of 10)
5 95 10 (0 to a maximum of 9)
10 90 10 (0 to a maximum of 9)
...
...
85 15 2 (0 to a maximum of 1)
90 10 2 (0 to a maximum of 1)
95 5 1 (only 0)
100 0 1 (only 0)
Add the numbers in column 3 to find the total number of ways of making change for $100 using $10, $5, and $1 bills.
Note there is a nice pattern in the numbers in column 3 that makes it possible to find the total without adding the numbers one at a time....