The vending machine price for potato chips is $1.50 per bag. The vending machine accepts nickels, dimes, quarters, loonies, and toonies. (In Canada, the $1 coin is called the loonie, and the $2 coin is called the toonie.)
How many ways can you pay $1.50 exactly?
If you pay with a toonie, how many ways can the machine give you your $0.50 change?
The government starts pushing the half-dollar coin. The vending machine gets modified to accept and dispense half-dollar coins. Answer each of the above cases with this change in effect.
(In all cases, for counting the number of ways, the order of the coins does not matter. It is only the number of each coin type that counts.)
Submit your answer to Gene Wirchenko <email@example.com>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, February 5, 2014 at noon Pacific Time. I will post the answer shortly after.