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You could choose 1 bill from the five given bills, each one obviously give you different sums. (1C5)
Then you could also choose 2 bills from the five given bills, this will give you different sums from each other and from what you have in step one. (2C5)
Then you could choose 3 bills from the five ...
So on and so forth
Until you choose all of them adding together, that is another different sum from the previous one. (5C5)
So total possible ways is 1C5+2C5+3C5+4C5+5C5=2(1C5+2C5)+1=2(5+5*4/2)+1=31