making_change.py

from typing import List

cache = {}

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.

    For instance, there are two ways to make a value of 3: with 3x 1 coins, or with 1x 1 coin and 1x 2 coin.
    """
    # Clear cache before starting new calculation
    cache.clear()
    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1])


def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
    """
    Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
    """
    # We use tuple(coins) because lists cannot be keys.
    key = (total, tuple(coins))

    if key in cache:
        return cache[key]
    
    if total == 0 or len(coins) == 0:
        return 0

    # Check if only one coin type remains
    if len(coins) == 1:
        if total % coins[0] == 0:
            return 1
        else:
            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:])
                ways += intermediate
            count_of_coin += 1
            
    # Store result in cache
    cache[key] = ways
    return ways