Add manual caching to ways_to_make_change to speed up recursive calls

Author: sheida-shab

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -10,13 +10,18 @@ def ways_to_make_change(total: int) -> int:
     return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1])
 
 
-def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
+def ways_to_make_change_helper(total: int, coins: List[int],cache=None) -> 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
-
+    if cache is None:
+        cache={}
+    key=(total,tuple(coins))    
+    if key in cache:
+        return cache[key]
+    
     ways = 0
     for coin_index in range(len(coins)):
         coin = coins[coin_index]
@@ -26,7 +31,8 @@ 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
+    cache[key]=ways        
     return ways