making_change.py

from typing import List

# Cache dictionary to store computed results
# Key: (total, tuple(coins)) -> Value: number of ways
change_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 for each new call to avoid interference between tests
    change_cache.clear()
    coins = [200, 100, 50, 20, 10, 5, 2, 1]
    return ways_to_make_change_helper(total, coins)


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.
    """
    # Create a cache key - use tuple of coins since lists are not hashable
    cache_key = (total, tuple(coins))
    
    # Check if we already computed this combination
    if cache_key in change_cache:
        return change_cache[cache_key]

    # Base cases
    if total == 0:
        return 1  # One way to make 0: use no coins
    if len(coins) == 0:
        return 0  # No coins left but still need to make total > 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 before returning
    change_cache[cache_key] = ways
    return ways