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.


let changes amount coins =
    let ways = Array.zeroCreate (amount + 1)
    ways.[0] <- 1L
    List.iter (fun coin ->
        for j = coin to amount do ways.[j] <- ways.[j] + ways.[j - coin]
    ) coins
    ways.[amount]
 
[<EntryPoint>]
let main argv = 
    printfn "%d" (changes    100 [25; 10; 5; 1]);
    printfn "%d" (changes 100000 [100; 50; 25; 10; 5; 1]);
    0