A set of coins makes change for n if the sum of the values of the coins is n. For example, if you have 1-cent, 2-cent and 4-cent coins, the following sets make change for 7:
- 7 1-cent coins
- 5 1-cent, 1 2-cent coins
- 3 1-cent, 2 2-cent coins
- 3 1-cent, 1 4-cent coins
- 1 1-cent, 3 2-cent coins
- 1 1-cent, 1 2-cent, 1 4-cent coins
Thus, there are 6 ways to make change for 7. This function count_changetakes a positive integer n and a list of the coin denominations and returns the number of ways to make change for n using these coins (Hint: You will need to use tree recursion):
def count_change(amount, denominations):
"""Returns the number of ways to make change for amount.
>>> denominations = [50, 25, 10, 5, 1]
>>> count_change(7, denominations)
2
>>> count_change(100, denominations)
292
>>> denominations = [16, 8, 4, 2, 1]
>>> count_change(7, denominations)
6
>>> count_change(10, denominations)
14
>>> count_change(20, denominations)
60
"""
"*** YOUR CODE HERE ***"
def count_using(min_coin, amount):
if amount < 0:
return 0
elif amount == 0:
return 1
elif min_coin > amount:
return 0
else:
with_min = count_using(min_coin, amount - min_coin)
without_min = count_using(2*min_coin, amount)
return with_min + without_min
return count_using(1, amount)