making_change.py

from typing import List

def ways_to_make_change(total: int) -> int:
    """
    Given access to coins with the values 1, 2, 5, 10, 20, 50, 100, 200,
    returns a count of all of the ways to make the passed total value.
    """
    cache = {}
    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1], cache)


def ways_to_make_change_helper(total: int, coins: List[int], cache: dict) -> int:
    """
    Helper function with memoization.
    Cache key is (total, index of first coin in list) — but since we pass
    the coins list by slicing, we can cache using tuple(total, tuple(coins)).
    """

    key = (total, tuple(coins))

    if key in cache:
        return cache[key]

    if total == 0 or len(coins) == 0:
        return 0

    ways = 0
    for coin_index in range(len(coins)):
        coin = coins[coin_index]
        count_of_coin = 1
        while coin * count_of_coin <= total:
            total_from_coins = coin * count_of_coin
            if total_from_coins == total:
                ways += 1
            else:
                intermediate = ways_to_make_change_helper(
                    total - total_from_coins,
                    coins=coins[coin_index + 1:],
                    cache=cache
                )
                ways += intermediate
            count_of_coin += 1

    cache[key] = ways
    return ways