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.


 
Module FindCoins {
      Function count(c(), n)  {
            dim table(n+1)=0@ :  table(0)=1@
            for c=0 to len(c())-1 {
                 if c(c)>n then exit
            }
            if c else exit
            for i=0 to c-1 {for j=c(i) to n {table(j)+=table(j-c(i))}}
            =table(n)
      }
      Print "For 1$ ways to change:";count((1,5,10,25),100)
      Print "For 100$ (optional task ways to change):";count((1,5,10,25,50,100),100000)
}
FindCoins
 