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.


package main
 
import "fmt"
 
func main() {
    amount := 100
    fmt.Println("amount, ways to make change:", amount, countChange(amount))
}
 
func countChange(amount int) int64 {
    return cc(amount, 4)
}
 
func cc(amount, kindsOfCoins int) int64 {
    switch {
    case amount == 0:
        return 1
    case amount < 0 || kindsOfCoins == 0:
        return 0
    }
    return cc(amount, kindsOfCoins-1) +
        cc(amount - firstDenomination(kindsOfCoins), kindsOfCoins)
}
 
func firstDenomination(kindsOfCoins int) int {
    switch kindsOfCoins {
    case 1:
        return 1
    case 2:
        return 5
    case 3:
        return 10
    case 4:
        return 25
    }
    panic(kindsOfCoins)
}