Probability of a Two Boxes Having The Same Number of Distinc

/*
Problem Statement: https://leetcode.com/problems/probability-of-a-two-boxes-having-the-same-number-of-distinct-balls/
*/

class Solution {
public:
    double getProbability(vector<int>& balls) {
        int k, n;
        double valid, total;
        k = balls.size();
        n = accumulate(balls.begin(), balls.end(), 0) / 2;
        valid = total = 0;
        vector<int> b(k);

        // helper functions
        auto fact = [&](int n) {
            double f = 1;
            while (n)
                f *= n--;
            return f;
        };
        auto state = [&]() {
            int box_1, box_2;
            double f1, f2;
            if (accumulate(b.begin(), b.end(), 0) != n)
                return;
            box_1 = box_2 = 0;
            f1 = f2 = fact(n);
            for (int i = 0; i < k; i++) {
                box_1 += b[i] > 0;
                box_2 += balls[i] - b[i] > 0;
                if (b[i]) {
                    box_1++;
                    f1 /= fact(b[i]);
                }
                if (balls[i] - b[i]) {
                    box_2++;
                    f2 /= fact(balls[i] - b[i]);
                }
            }
            valid += (box_1 == box_2) * f1 * f2;
            total += f1 * f2;
        };
        function<void(int)> recurse = [&](int pos) {
            if (pos == k) {
                state();
                return;
            }
            for (int i = 0; i <= balls[pos]; i++) {
                b[pos] = i;
                recurse(pos + 1);
            }
        };
        recurse(0);

        return valid / total;
    }
};