Finding all possible combinations of numbers to reach a given sum

Question

How would you go about testing all possible combinations of additions from a given set N of numbers so they add up to a given final number?

A brief exam

Answer 1

In Haskell:

filter ((==) 12345 . sum) $ subsequences [1,5,22,15,0,..]

And J:

(]#~12345=+/@>)(]<@#~[:#:@i.2^#)1 5 22 15 0 ...

As you may notice, both take the same approach and divide the problem into two parts: generate each member of the power set, and check each member's sum to the target.

There are other solutions but this is the most straightforward.

Do you need help with either one, or finding a different approach?