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);
        valid = total = 0;
        vector<int> b(k);
        double nCr[n + 1][n + 1];

        // initialize nCr
        for (int i = 0; i < n; i++)
            nCr[i][0] = 1;
        for (int i = 1; i < n; i++)
            nCr[0][i] = 0;
        for (int i = 1; i < n; i++)
            for (int j = 1; j < n; j++)
                nCr[i][j] = nCr[i - 1][j - 1] + nCr[i - 1][j];

        // helper function
        auto state = [&]() {
            if (accumulate(b.begin(), b.end(), 0) != n / 2)
                    return;
            double ways = 1;
            int box_1, box_2;
            box_1 = box_2 = 0;

            for (int i = 0; i < k; i++) {
                box_1 += b[i] > 0;
                box_2 += balls[i] - b[i] > 0;
                ways *= nCr[balls[i]][b[i]];
            }

            valid += (box_1 == box_2) * ways;
            total += ways;
        };
        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;
    }
};