The subset-sum problem is to find a subset of a set of integers that sums to a given value. The decision problem of finding out if such a subset exists is NP-complete. One way of solving the problem is to use backtracking. This is a backtracking solution in C that finds all of the subsets that sum to the target value.

typedef void(*subset_sumfn)(const unsigned int *, size_t);

static unsigned int promising(int i, size_t len, unsigned int weight, unsigned int total, 
        unsigned int target, const unsigned int *weights)
{
    return (weight + total >= target) && (weight == target || weight + weights[i + 1] <= target);
}

static unsigned int sum(const unsigned int *weights, size_t len)
{
    unsigned int total = 0;
    unsigned int i;
    for (i = 0; i < len; i++) {
        total += weights[i];
    }
    return total;
}

static void subset_sum_recursive(const unsigned int *weights, size_t len, unsigned int target, 
        int i, unsigned int weight, unsigned int total, unsigned int *include, subset_sumfn fun)
{
    if (promising(i, len, weight, total, target, weights)) {
        if (weight == target) {
            fun(include, i + 1);
        }
        else if (i < (int)len - 1){
            include[i + 1] = 1;
            subset_sum_recursive(weights, len, target, i + 1, weight + weights[i + 1], 
                   total - weights[i + 1], include, fun);
            include[i + 1] = 0;
            subset_sum_recursive(weights, len, target, i + 1, weight,
                    total - weights[i + 1], include, fun);
        }
    }
}

void subset_sum(const unsigned int *weights, size_t len, unsigned int target, subset_sumfn fun)
{
    const unsigned int total = sum(weights, len);
    unsigned int *include = calloc(len, sizeof(unsigned int));
    if (include == NULL) {
        return;
    }
    subset_sum_recursive(weights, len, target, -1, 0, total, include, fun);
    free(include);
}

int main(void)
{
    unsigned int weights[] = {1, 2, 3, 4, 5, 6, 7, 8, 9};
    const unsigned int len = sizeof(weights) / sizeof(unsigned int);
    const unsigned int target = 7;
    subset_sum(weights, len, target, print_vector);
    return 0;
}

The output is in the form of binary strings showing which elements of the original set are in the subset, so for example the first binary string corresponds to 1 + 2 + 4 = 7.

1 1 0 1
1 0 0 0 0 1
0 1 0 0 1
0 0 1 1
0 0 0 0 0 0 1