Problem

import java.util.ArrayList;
import java.util.Collections;
import java.util.List;

public final class Problem
{
    static class IntList
    {
        IntList (int i, IntList next) {
            this.i = i;
            this.next = next;
        }

        public int i;
        public IntList next;
    }

    // works best if set sorted in increasing order
    static void solutions (int target, int... set)
    {
        IntList remaining = null;
        for (int i: set)
            remaining = new IntList(i, remaining);

        List<IntList> solutions = solutions(target, 0, null, remaining);

        int i = 1;
        for (IntList solution: solutions) {
            System.out.print("solution " + i + ": ");
            while (solution != null) {
                System.out.print(solution.i);
                if (solution.next != null) System.out.print(", ");
                solution = solution.next;
            }
            System.out.println();
            ++i;
        }
    }

    static List<IntList> solutions (int target, int total, IntList selected, IntList remaining)
    {
        if (remaining == null)
            return Collections.emptyList();

        List<IntList> solutions = new ArrayList<>();
        int total2 = total + remaining.i;

        if (total2 == target)
            // first item yield a solution
            solutions.add(new IntList(remaining.i, selected));
        else if (total2 < target)
            // include first item
            solutions.addAll(
                solutions(target, total2, new IntList(remaining.i, selected), remaining.next));

        // omit first item
        solutions.addAll(solutions(target, total, selected, remaining.next));
        return solutions;
    }

    public static void main (String[] args)
    {
        solutions(7, 1, 2, 3, 4);
    }
}