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

@KeithBeller's answer with slightly changed variable names and some comments.

    public static void Main(string[] args)
    {
        List input = new List() { 3, 9, 8, 4, 5, 7, 10 };
        int targetSum = 15;
        SumUp(input, targetSum);
    }

    public static void SumUp(List input, int targetSum)
    {
        SumUpRecursive(input, targetSum, new List());
    }

    private static void SumUpRecursive(List remaining, int targetSum, List listToSum)
    {
        // Sum up partial
        int sum = 0;
        foreach (int x in listToSum)
            sum += x;

        //Check sum matched
        if (sum == targetSum)
            Console.WriteLine("sum(" + string.Join(",", listToSum.ToArray()) + ")=" + targetSum);

        //Check sum passed
        if (sum >= targetSum)
            return;

        //Iterate each input character
        for (int i = 0; i < remaining.Count; i++)
        {
            //Build list of remaining items to iterate
            List newRemaining = new List();
            for (int j = i + 1; j < remaining.Count; j++)
                newRemaining.Add(remaining[j]);

            //Update partial list
            List newListToSum = new List(listToSum);
            int currentItem = remaining[i];
            newListToSum.Add(currentItem);
            SumUpRecursive(newRemaining, targetSum, newListToSum);
        }
    }'