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.


(def denomination-kind [1 5 10 25])
 
(defn- cc [amount denominations]
  (cond (= amount 0) 1
        (or (< amount 0) (empty? denominations)) 0
        :else (+ (cc amount (rest denominations))
                 (cc (- amount (first denominations)) denominations))))
 
(defn count-change
  "Calculates the number of times you can give change with the given denominations."
  [amount denominations]
  (cc amount denominations))
 
(count-change 15 denomination-kind) ; = 6 