#math trivia for #September7: #250 cents can be “changed” into dollars and quarters how many ways? How about dollars, quarters and dimes?

— Burt Kaliski Jr. (@modulomathy) September 7, 2013

There are three ways to change 250 cents or $2.50 into dollars and quarters, depending on the number of dollars given:

- Two dollars, two quarters
- One dollar, six quarters
- No dollars, 10 quarters

To include dimes, we can start with the number of dollars and then break up the rest of the change into quarters and dimes in various ways, exchanging each successive set of two quarters for five dimes:

- Two dollars AND
- Two quarters OR
- Five dimes

- One dollar AND
- Six quarters OR
- Four quarters, five dimes OR
- Two quarters, 10 dimes OR
- 15 dimes

- No dollars AND
- 10 quarters OR
- Eight quarters, five dimes OR
- Six quarters, 10 dimes OR
- Four quarters, 15 dimes OR
- Two quarters, 20 dimes OR
- 25 dimes

This adds up to 12 ways.

For the reader: What if nickels were also allowed? Pennies too?