Puzzle #227: Frozen Yummy

The yummy, chocolate-coated, ice cream thing on a stick that you – OK, OK, I – have been buying costs $4.20 at the local store near where I work.

How many ways are there to pay $4.20 with exact change using the usual Canadian coin denominations (nickel, dime, quarter, loonie, and toonie)? Wait; that would be boring. How many combinations consist of six or fewer coins?

[Hint: You can save yourself some effort if you look at the problem in the right way.]

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, October 11, 2017 at noon Pacific Time. I will post the answer shortly after.