Refactor ways_to_make_change to use memoization

Author: eyuell21

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -1,32 +1,41 @@
-from typing import List
-
+from typing import List, Dict, Tuple
 
 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.
     """
-    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1])
+    coins = [200, 100, 50, 20, 10, 5, 2, 1]
+    memo: Dict[Tuple[int, int], int] = {}
+    return ways_to_make_change_helper(total, coins, memo)
 
 
-def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
+def ways_to_make_change_helper(total: int, coins: List[int], memo: Dict[Tuple[int, int], int]) -> int:
     """
     Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
+    Uses memoization to cache results for (total, coin_index).
     """
-    if total == 0 or len(coins) == 0:
-        return 0
+    if total == 0:
+        return 1  
+    if total < 0 or not coins:
+        return 0  
+
+    key = (total, len(coins))
+    if key in memo:
+        return memo[key]
 
     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:
+            remainder = total - coin * count_of_coin
+            if remainder == 0:
                 ways += 1
             else:
-                intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:])
-                ways += intermediate
+                ways += ways_to_make_change_helper(remainder, coins[coin_index + 1:], memo)
             count_of_coin += 1
+
+    memo[key] = ways
     return ways