Improve making change caching

Author: AliQassab

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -7,16 +7,24 @@ 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])
+    # Cache to store computed results
+    cache = {}
+    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1], cache)
 
 
-def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
+def ways_to_make_change_helper(total: int, coins: List[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:
         return 0
 
+    # Create cache key from current state
+    cache_key = (total, tuple(coins))
+    
+    if cache_key in cache:
+        return cache[cache_key]
+
     ways = 0
     for coin_index in range(len(coins)):
         coin = coins[coin_index]
@@ -26,7 +34,10 @@ def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
             if total_from_coins == total:
                 ways += 1
             else:
-                intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:])
+                intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:], cache=cache)
                 ways += intermediate
             count_of_coin += 1
+    
+    # Store result in cache
+    cache[cache_key] = ways
     return ways