There are four types of common coins in   US   currency:

   quarters   (25 cents)
   dimes   (10 cents)
   nickels   (5 cents),   and 
   pennies   (1 cent) 


There are six ways to make change for 15 cents:

   A dime and a nickel 
   A dime and 5 pennies
   3 nickels
   2 nickels and 5 pennies
   A nickel and 10 pennies
   15 pennies


Task

How many ways are there to make change for a dollar using these common coins?     (1 dollar = 100 cents).



Optional

Less common are dollar coins (100 cents);   and very rare are half dollars (50 cents).   With the addition of these two coins, how many ways are there to make change for $1000?

(Note:   the answer is larger than   232).



References

 an algorithm from the book Structure and Interpretation of Computer Programs.
 an article in the algorithmist.
 Change-making problem on Wikipedia.


 
templates makeChange&{coins:}
  def paid: $;
  @: [1..$paid -> 0];
  $coins... -> \(def coin: $;
    @makeChange($coin): $@makeChange($coin) + 1;
    $coin+1..$paid -> @makeChange($): $@makeChange($) + $@makeChange($-$coin);
  \) -> !VOID
  $@($paid)!
end makeChange
 
100 -> makeChange&{coins: [1,5,10,25]} -> '$; ways to change a dollar
' -> !OUT::write
100000 -> makeChange&{coins: [1,5,10,25,50,100]} -> '$; ways to change 1000 dollars with all coins
' -> !OUT::write
 