Question 1119006
<br>
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.<br>
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.<br>
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.<br>
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.<br>
So...<br><pre>
  # 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)</pre><br>
The total number of ways is then seen to be<br>
2(1+2+...+31) = 2((31*32)/2) = 992