How many ways are there to add the numbers in set $k$ to equal $n$? How many ways are there to add the numbers in set $k$ to equal $n$?
For a specific example, consider the following: I have infinite pennies, nickels, dimes, quarters, and loonies (equivalent to 0.01, 0.05, 0.1, 0.25, and 1, for those who are not Canadian). How many unique ways are there to add these numbers to get a loonie?
Some combinations would include:
$1 = 1$
$1 = 0.25 + 0.25 + 0.25 + 0.25$
$1 = 0.25 + 0.25 + 0.25 + 0.1 + 0.1 + 0.05$
And so on.
Several people have suggested the coin problem, though I have yet to find a source that explains this well enough for me to understand it.

You will have luck googling for this with the phrase "coin problem". 
I have a few links at this solution which will lead to the general method. There is a Project Euler problem (or maybe several of them) which ask you to compute absurdly large such numbers of ways, but the programs (I found out) can be just a handful of lines.
