  Given a dollar, how many ways are there to make change using pennies,
nickels, dimes, and quarters? What if you are given a different amount of
money? What if you use different coin denominations? This is a well known
problem and formulas are known. We present simpler proofs in several cases. We
use recurrences to derive formulas if the coin denominations are {1,x,kx,rx},
and we use a simple proof using generating functions to derive a formula for
any coin set.
