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how many different sums of money can you make using one or more of these bills?

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The trick to solving this problem is realizing that you need to break it up into parts.  Find the number of totals with one, two, three, or four bills, then add them up.  Since this is a combination problem, you can find each of the numbers with the combination formula, but the numbers are small enough to just think this one out.
One Bill: 4 possibilities
Two Bills: 6 possibilities
Three Bills: 4 possibilities
Four Bills: 1 possibility
4+6+4+1=15 total different sums of money