Hi everyone - new here!

So I've been an given an extension problem by a professor and have been trying to figure it out intuitively but cannot construct a solution...it is based on the lottery system in Antarctica. Help appreciated!

Q:

In the Antarctica lottery, there are 38 balls, with 6 numbers being drawn each week. The 2 types of tickets are as follows:

a) Regular Entry - this is a ticket where you just select 6 numbers per game - a regular lotto entry

b) System Entry - on this ticket, you can select between 7-20 numbers and your entry will automatically cover all combinations of every number on the ticket

The System Entry ticket type will allow you to select up to 20 numbers, and charge you for all combinations accordingly. Unfortunately, for Mr Penguin, who loves to gamble, he wishes to select more than 20 numbers.

For example, this week, for Christmas, Mr Penguin wishes to select every combination of every number between 1-25. What is the most efficient way he can do this? That is, how many different tickets/ticket types will Mr Penguin have to purchase and how much will it cost him? Can you construct a simple table outlining this for each of the numbers >20

See attached spreadsheet for further explanation of ticket prices/combinations.

All help very much appreciated - will award a gold star