Clearly, using only 20s, 10s, and 1s, the number of 1s to make a total of $614 must be a number with units digit 4. So make an organized list of the different possible numbers of $1 bills; for each of those numbers, find the number of combinations of 20s and 10s that make the remainder of the $614.
For each such case, that will be easy; using only 20s and 10s, the number of ways to make the remaining amount is just the number of choices you have for the number of 20s.
For example, if there are 104 $1 bills, the remaining amount is $510. The number of 20s you can have is any whole number for which the sum of the $20 bills is less than or equal to $510. That means 0 to 25 $20 bills, making 26 ways to make the $614 total using 104 $1 bills.
Then look for a pattern in the numbers of ways for the different cases to find an easy way to determine the total number of ways.
So...
# of $1 remaining # of
bills amount ways
-------------------------
4 610 31 (0 to 30)
14 600 31 (0 to 30)
24 590 30 (0 to 29)
34 580 30 (0 to 29)
44 570 29
...
584 30 2 (0 or 1)
594 20 2 (0 or 1)
604 10 1 (0)
614 0 1 (0)
The total number of ways is then seen to be
2(1+2+...+31) = 2((31*32)/2) = 992