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

Here is a better version with better output formatting and C++ 11 features:

void subset_sum_rec(std::vector & nums, const int & target, std::vector & partialNums) 
{
    int currentSum = std::accumulate(partialNums.begin(), partialNums.end(), 0);
    if (currentSum > target)
        return;
    if (currentSum == target) 
    {
        std::cout << "sum([";
        for (auto it = partialNums.begin(); it != std::prev(partialNums.end()); ++it)
            cout << *it << ",";
        cout << *std::prev(partialNums.end());
        std::cout << "])=" << target << std::endl;
    }
    for (auto it = nums.begin(); it != nums.end(); ++it) 
    {
        std::vector remaining;
        for (auto it2 = std::next(it); it2 != nums.end(); ++it2)
            remaining.push_back(*it2);

        std::vector partial = partialNums;
        partial.push_back(*it);
        subset_sum_rec(remaining, target, partial);
    }
}