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

C# version of @msalvadores code answer

void Main()
{
    int[] numbers = {3,9,8,4,5,7,10};
    int target = 15;
    sum_up(new List(numbers.ToList()),target);
}

static void sum_up_recursive(List numbers, int target, List part)
{
   int s = 0;
   foreach (int x in part)
   {
       s += x;
   }
   if (s == target)
   {
        Console.WriteLine("sum(" + string.Join(",", part.Select(n => n.ToString()).ToArray()) + ")=" + target);
   }
   if (s >= target)
   {
        return;
   }
   for (int i = 0;i < numbers.Count;i++)
   {
         var remaining = new List();
         int n = numbers[i];
         for (int j = i + 1; j < numbers.Count;j++)
         {
             remaining.Add(numbers[j]);
         }
         var part_rec = new List(part);
         part_rec.Add(n);
         sum_up_recursive(remaining,target,part_rec);
   }
}
static void sum_up(List numbers, int target)
{
    sum_up_recursive(numbers,target,new List());
}