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.


import std.stdio, std.bigint;
 
auto changes(int amount, int[] coins) {
    auto ways = new BigInt[amount + 1];
    ways[0] = 1;
    foreach (coin; coins)
        foreach (j; coin .. amount + 1)
            ways[j] += ways[j - coin];
    return ways[$ - 1];
}
 
void main() {
    changes(   1_00, [25, 10, 5, 1]).writeln;
    changes(1000_00, [100, 50, 25, 10, 5, 1]).writeln;
}