Ways to Break a Dollar into Change

first_attempt.py

def ways_to_break(amount, coins):
    """
    :param amount: the monetary value to break into some number of coins
    :param coins: a container of descending coin denominations
    :return: the number of different ways to break amount into coins
    """
    this_coin = coins[0]

    # If this is the only coin, there's one way to break it.
    if len(coins) == 1:
        return 1

    # Note range is never empty, we always want to try with 0 of this coin.
    total = 0
    for coin_value in range(0, amount+1, this_coin):
        total += ways_to_break(amount - coin_value, coins[1:])
    return total


print ways_to_break(100, (100, 50, 25, 10, 5, 1))

works_in_more_cases.py

def ways_to_break(amount, coins):
    """
    :param amount: the monetary value to break into some number of coins
    :param coins: a mutable container of coin denominations
    :return: the number of different ways to break amount into coins
    """
    this_coin = coins.pop()

    # If that was the only coin, see if it can break amount.
    if not coins:
        if amount % this_coin:
            return 0
        else:
            return 1

    # For every time this coin denomination could be used (including zero),
    # get the number of ways to break the remaining amount with only the 
    # remaining coins.
    return sum([ways_to_break(amount - coin_value, coins[:])
                for coin_value in range(0, amount+1, this_coin)])


# The list of coins doesn't have to be ordered now, but there'll
# be fewer calls if the biggest denominations are processed first.
print ways_to_break(100, [1, 5, 10, 25, 50, 100])