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> I think this is known as the Subset Sum Problem.
Well ... almost. The Subset Sum Problem asks if there is one solution. But the OP asks to "list all possible combination of numbers" Algorithm wise that's a huge difference, because one can often optimize searching for a single solution, while happily ignoring the rest. ° And that's also why I'm hesitant solving this, you can easily show that the solution space of all possible combinations will explode quickly, in a way that already the time needed to print them out will take an eternity. I.O.W. such problems don't make much sense, unless you are singling out a single (or a few) solution which are optimal regarding a second valuefunction.
Cheers Rolf °)The Knapsack Problem will also demand to optimize a second "value" function and only require the "weight" to be less or equal the "target". It's a generalization of Subset Sum b/c if you choose the weight as value, the equal case  if it exists  will be maximal. In reply to Re^2: Find combination of numbers whose sum equals X
by LanX

