refactor: optimize making_change to use tuple with index parameter

Author: AliQassab

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -1,43 +1,38 @@
-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.
 
     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.
     """
-    # Cache to store computed results
     cache = {}
-    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1], cache)
+    coins = (200, 100, 50, 20, 10, 5, 2, 1)
+    return ways_to_make_change_helper(total, coins, 0, cache)
 
 
-def ways_to_make_change_helper(total: int, coins: List[int], cache: dict) -> int:
+def ways_to_make_change_helper(total: int, coins: tuple, coin_index: int, cache: dict) -> int:
     """
     Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
     """
-    if total == 0 or len(coins) == 0:
+    if total == 0 or coin_index >= len(coins):
         return 0
 
-    # Create cache key from current state
-    cache_key = (total, tuple(coins))
+    cache_key = (total, coin_index)
 
     if cache_key in cache:
         return cache[cache_key]
 
     ways = 0
-    for coin_index in range(len(coins)):
-        coin = coins[coin_index]
+    for i in range(coin_index, len(coins)):
+        coin = coins[i]
         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)
+                intermediate = ways_to_make_change_helper(total - total_from_coins, coins, i + 1, cache)
                 ways += intermediate
             count_of_coin += 1
 
-    # Store result in cache
     cache[cache_key] = ways
     return ways