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+#!/usr/bin/env python
+
+from functools import cache
+
+# establish the values of the coins
+types = { 50: 'HD', 25: 'Q', 10: 'D', 5: 'N', 1: 'P' }
+
+# make a sorted list of coin values
+values = sorted(types.keys(), reverse=True)
+
+def formatCoin(coin, count):
+ return f"{count}{coin}" if count > 1 else coin
+
+@cache
+def makeChange(amount, exclude = 0):
+ output = []
+ for value in values:
+ # if this type of coin is worth more
+ # than the total, we can't use it
+ if value > amount:
+ continue
+
+ # if we're excluding this coin value or greater, skip it
+ if exclude and value >= exclude:
+ continue
+
+ coin = types[value] # get the coin name
+
+ if value == 1:
+ # pennies are easy!
+ output.append( formatCoin(coin, amount) )
+ else:
+ # loop from max number of this coin we can use down to 1
+ for count in range(int(amount / value), 0, -1):
+ # figure out how much change we need after we've used
+ # $count many of coin $coin
+ left = amount - (count * value)
+
+ if left > 0: # we need more change
+ # make a recursive call to get the combinations
+ # for that remaining amount, excluding any coins
+ # of equal or greater value to $value
+ combinations = makeChange(left, value)
+ for combo in combinations:
+ this_coin = formatCoin(coin, count)
+ output.append(" + ".join([this_coin, combo]))
+ else: # this is exact change!
+ output.append( formatCoin(coin, count) )
+ return output
+
+def solution(amount):
+ print(f'Input: $amount = {amount}')
+ ways = makeChange(amount)
+ print(f'Output: {len(ways)}')
+ print('')
+ i = 0
+ for c in ways:
+ i += 1
+ print(f'{i}: {c}')
+
+print('Example 1:')
+solution(9)
+
+print('\nExample 2:')
+solution(15)
+
+print('\nExample 3:')
+solution(100)