add memoization to making_change

Author: katarzynakaz

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -7,26 +7,35 @@ def ways_to_make_change(total: int) -> int:
 
     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 = {}
+    return ways_to_make_change_helper(total, coins, 0, memo)
 
-
-def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
+def ways_to_make_change_helper(total: int, coins: List[int], coin_index: int, memo: 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:
+        return 1
+
+    if coin_index == len(coins):
         return 0
+    
+    key = (total, coin_index)
+
+    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:
-                ways += 1
-            else:
-                intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:])
-                ways += intermediate
-            count_of_coin += 1
+    coin = coins[coin_index]
+    count_of_coin = 0
+
+    while count_of_coin * coin <= total:
+        remaining = total - count_of_coin * coin
+        ways += ways_to_make_change_helper(remaining, coins, coin_index + 1, memo)
+        count_of_coin += 1
+    
+    memo[key] = ways
+
     return ways