#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

struct Item {
    int basePower;
    vector<int> bonusPower;
};

bool compareItems(const Item& item1, const Item& item2) {
    return item1.bonusPower.back() > item2.bonusPower.back();
}

int main() {
    int n, k;
    cin >> n >> k;

    vector<Item> items(n);
    for (int i = 0; i < n; i++) {
        int p;
        cin >> p;
        items[i].basePower = p;
        items[i].bonusPower.resize(p);
        for (int j = 0; j < p; j++) {
            cin >> items[i].bonusPower[j];
        }
        sort(items[i].bonusPower.begin(), items[i].bonusPower.end());
    }

    sort(items.begin(), items.end(), compareItems);

    int totalBonusPower = 0;
    int sum = 0;
    for (int i = 0; i < n; i++) {
        if (sum + items[i].basePower <= k) {
            totalBonusPower += items[i].bonusPower.back();
            sum += items[i].basePower;
        } else if (sum < k) {
            int remainingPower = k - sum;
            totalBonusPower += items[i].bonusPower[items[i].basePower - remainingPower];
            sum = k;
        } else {
            break;
        }
    }

    cout << totalBonusPower << endl;

    return 0;
}