How to count the combinations not greater than a given volume in a knapsack problem?

Question

You are given a certain set of items with weight wi and the volume of the bag.

Case:

Input: 

3 (count of items) 10 (volume of the bag)
1 2 4 (weight for each item)

Output

8 (since you can put any item in or not 2 * 2 *2)

I have a naive solution using DFS as

public class Main {
    public static void main(String... args) throws Exception {
        Scanner scanner = new Scanner(System.in);
        int size = scanner.nextInt();
        int volume = scanner.nextInt();
        int[] weights = new int[size];
        for (int i = 0; i < size; ++i) weights[i] = scanner.nextInt();
        System.out.println(getCounts(weights, 0, volume));
    }

    private static int getCounts(int[] nums, int pos, int target) {
        if (target < 0) return 0;
        if (pos >= nums.length) return 1;
        return getCounts(nums, pos+1, target) +
            getCounts(nums, pos+1, target-nums[pos]);
    }
}

As expected, it produced TLE and I added cache to avoid re-calculation as

public class Main {
    private static int[][] cache;
    public static void main(String... args) throws Exception {
        Scanner scanner = new Scanner(System.in);
        int size = scanner.nextInt();
        int volume = scanner.nextInt();
        cache = new int[size+1][volume+1];
        for (int i = 0; i <= size; ++i) {
            for (int j = 0; j <= volume; ++j) {
                cache[i][j] = -1;
            }
        }
        int[] weights = new int[size];
        for (int i = 0; i < size; ++i) weights[i] = scanner.nextInt();
        System.out.println(getCounts(weights, 0, volume));
    }

    private static int getCounts(int[] nums, int pos, int target) {
        if (target < 0) return 0;
        if (pos == nums.length) return 1;
        if (cache[pos][target] != -1) return cache[pos][target];
        int count = getCounts(nums, pos+1, target) +
            getCounts(nums, pos+1, target-nums[pos]);
        cache[pos][target] = count;
        return count;
    }
}

But the OJ prompt me of memory error while the memory is within the limit => 13064K < 32768K, any idea to solve this issue?

1. Why did it fail?
2. Could a DP solution without backtracking be found for this?

Any help will be appreciated :)

Here is the original problem in Chinese.