From line 36, we can figure out coins can only be one of the following 9 lists:

[200, 100, 50, 20, 10, 5, 2, 1]
[100, 50, 20, 10, 5, 2, 1]
[50, 20, 10, 5, 2, 1]
...
[1]
[]

We could further improve the performance if we can

• avoid repeatedly creating the same sub-lists at line 36, and
• create key as (total, a_value_identifiying_one_of_the_sublists) instead of as (total, tuple(coins))

I don't think this exercise expects trainees to optimize the code to this level. So change is optional.