简单组合数学（二）

一、分割序列

import java.util.*;
public class Main {
    static final long MOD = 1000000007L;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        String s = sc.next();
        List<Integer> pos = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            if (s.charAt(i) == '1') {
                pos.add(i);
            }
        }
        if (pos.isEmpty()) {
            System.out.println(0);
            return;
        }
        long ans = 1;
        for (int i = 1; i < pos.size(); i++) {
            long diff = pos.get(i) - pos.get(i - 1);
            ans = ans * diff % MOD;
        }
        System.out.println(ans);
    }
}

运行结果

二、礼物清单

import java.util.Scanner;

public class Main {
    static final long MOD = 1000000007L;

    static long qpow(long a, long b, long mod) {
        long res = 1;
        a %= mod;
        while (b > 0) {
            if (b % 2 == 1) {
                res = res * a % mod;
            }
            a = a * a % mod;
            b >>= 1;
        }
        return res;
    }

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        long n = sc.nextLong();
        long m = sc.nextLong();

        long base = qpow(2, m, MOD);
        base = (base - 1 + MOD) % MOD;
        long ans = qpow(base, n, MOD);
        System.out.println(ans);
    }
}

运行结果

三、矩形计数

import java.util.Scanner;

public class Main {
    static final int MOD = 1000000007;
    static final int MAX = 1000;
    static long[] pow2 = new long[MAX + 1];

    static void pre() {
        pow2[0] = 1;
        for (int i = 1; i <= MAX; i++) {
            pow2[i] = pow2[i - 1] * 2 % MOD;
        }
    }

    public static void main(String[] args) {
        pre();
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        int m = sc.nextInt();
        int[][] a = new int[n][m];
        int[] row0 = new int[n];
        int[] row1 = new int[n];
        int[] col0 = new int[m];
        int[] col1 = new int[m];
        long totalSingle = 0;

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                a[i][j] = sc.nextInt();
                if (a[i][j] == 0) {
                    row0[i]++;
                    col0[j]++;
                } else {
                    row1[i]++;
                    col1[j]++;
                }
                totalSingle++;
            }
        }

        long sumRow = 0;
        for (int i = 0; i < n; i++) {
            long s0 = (pow2[row0[i]] - 1 + MOD) % MOD;
            long s1 = (pow2[row1[i]] - 1 + MOD) % MOD;
            sumRow = (sumRow + s0 + s1) % MOD;
        }
        long sumCol = 0;
        for (int j = 0; j < m; j++) {
            long s0 = (pow2[col0[j]] - 1 + MOD) % MOD;
            long s1 = (pow2[col1[j]] - 1 + MOD) % MOD;
            sumCol = (sumCol + s0 + s1) % MOD;
        }

        long ans = (sumRow + sumCol - totalSingle % MOD + MOD) % MOD;
        System.out.println(ans);
    }
}

运行结果

四、【模板】组合数

import java.io.*;
public class Main {
    static final int MOD = 1000000007;
    static final int MAX = 500000;
    static long[] fact = new long[MAX + 1];
    static long[] invfact = new long[MAX + 1];

    static long qpow(long a, long b) {
        long res = 1;
        while (b > 0) {
            if ((b & 1) == 1) res = res * a % MOD;
            a = a * a % MOD;
            b >>= 1;
        }
        return res;
    }

    static void pre() {
        fact[0] = 1;
        for (int i = 1; i <= MAX; i++) {
            fact[i] = fact[i - 1] * i % MOD;
        }
        invfact[MAX] = qpow(fact[MAX], MOD - 2);
        for (int i = MAX - 1; i >= 0; i--) {
            invfact[i] = invfact[i + 1] * (i + 1) % MOD;
        }
    }

    public static void main(String[] args) throws IOException {
        pre();
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        BufferedWriter bw = new BufferedWriter(new OutputStreamWriter(System.out));
        int T = Integer.parseInt(br.readLine());
        while (T-- > 0) {
            String[] s = br.readLine().split(" ");
            int n = Integer.parseInt(s[0]);
            int m = Integer.parseInt(s[1]);
            long ans = fact[m] * invfact[n] % MOD;
            ans = ans * invfact[m - n] % MOD;
            bw.write(ans + "\n");
        }
        bw.flush();
    }
}

运行结果

五、人员分组问题

import java.util.Scanner;

public class Main {
    static final long MOD = 1000000007L;
    static long inv120, inv720, inv5040;

    static long qpow(long a, long b) {
        long res = 1;
        while (b > 0) {
            if ((b & 1) == 1) res = res * a % MOD;
            a = a * a % MOD;
            b >>= 1;
        }
        return res;
    }

    static void pre() {
        inv120 = qpow(120, MOD - 2);
        inv720 = qpow(720, MOD - 2);
        inv5040 = qpow(5040, MOD - 2);
    }

    public static void main(String[] args) {
        pre();
        Scanner sc = new Scanner(System.in);
        long n = sc.nextLong();
        long c5 = 0, c6 = 0, c7 = 0;

        if (n >= 5) {
            c5 = n * (n - 1) % MOD;
            c5 = c5 * (n - 2) % MOD;
            c5 = c5 * (n - 3) % MOD;
            c5 = c5 * (n - 4) % MOD;
            c5 = c5 * inv120 % MOD;
        }
        if (n >= 6) {
            c6 = c5 * (n - 5) % MOD;
            c6 = c6 * 6 % MOD;
            c6 = c6 * inv720 % MOD;
        }
        if (n >= 7) {
            c7 = c6 * (n - 6) % MOD;
            c7 = c7 * 7 % MOD;
            c7 = c7 * inv5040 % MOD;
        }

        long ans = (c5 + c6 + c7) % MOD;
        System.out.println(ans);
    }
}

运行结果