每个小组中只有一位领导者的N个人中的小组数

给定N个人，任务是计算形成大小组的方式的数量? N ，在每个组中，组的第一个元素是组的领导者。
笔记：

• 具有相同领导者的人的组被视为不同的组。例如：组{1，2，3}和{2，1，3}被视为不同的组，因为它们分别具有不同的领导者1和2。
• 具有相同领导者且具有相同人员的组被视为同一组。例如：组{1、3、2}和{1、2、3}被视为同一组，因为它们具有相同的领导者和相同的人员。
• 答案可能非常大，取(1e9 + 7)为模。

例子：

Input: N = 3 
Output: 12 
Explanation: 
Total Groups with leaders are: 
Groups with Leader 1: 
1. {1} 
2. {1, 2} 
3. {1, 3} 
4. {1, 2, 3} 
Groups with Leader 2: 
5. {2} 
6. {2, 1} 
7. {2, 3} 
8. {2, 1, 3} 
Groups with Leader 3: 
9. {3} 
10. {3, 1} 
11. {3, 2} 
12. {3, 1, 2}

Input: N = 5 
Output: 80

方法：可以使用二项式系数和模幂的概念来解决此问题。下面是对该问题陈述的观察：

• 在N人中选择一位领导者的方式有C(N，1)个。
• 对于每个领导者，我们都可以选择大小为K的一组，其中0≤K≤N-1，以进行可能的分组数量。
• 因此，总数的方式由N与其余(N – 1)个元素中选择K个元素之和的乘积给出：

  Total Ways = $\binom{N}{1} * \left ( \binom{N-1}{0} + \binom{N-1}{1} + \binom{N-1}{2} + ... + \binom{N-1}{N-1} \right )$

通过使用二项式定理，二项式系数的总和可以写为：

$\binom{N}{1} * \left ( \binom{N-1}{0} + \binom{N-1}{1} + \binom{N-1}{2} + ... + \binom{N-1}{N-1} \right ) = 2^{N-1}$

因此，选择只有一位领导者的小组的方式是

$N * 2^{N - 1}$

下面是上述方法的实现：

C++

// C++ program for the above approach
  
#include 
using namespace std;
  
long long mod = 1000000007;
  
// Function to find 2^x using
// modular exponentiation
int exponentMod(int A, int B)
{
    // Base cases
    if (A == 0)
        return 0;
    if (B == 0)
        return 1;
  
    // If B is even
    long long y;
    if (B % 2 == 0) {
        y = exponentMod(A, B / 2);
        y = (y * y) % mod;
    }
  
    // If B is odd
    else {
        y = A % mod;
        y = (y * exponentMod(A, B - 1)
             % mod)
            % mod;
    }
  
    return (int)((y + mod) % mod);
}
  
// Function to count the number of
// ways to form the group having
// one leader
void countWays(int N)
{
  
    // Find 2^(N-1) using modular
    // exponentiation
    long long select = exponentMod(2,
                                   N - 1);
  
    // Count total ways
    long long ways
        = ((N % mod)
           * (select % mod));
  
    ways %= mod;
  
    // Print the total ways
    cout << ways;
}
  
// Driver Code
int main()
{
  
    // Given N number of peoples
    int N = 5;
  
    // Function Call
    countWays(N);
}

Java

// Java program for the above approach
import java.util.*;
class GFG{
  
static long mod = 1000000007;
  
// Function to find 2^x using
// modular exponentiation
static int exponentMod(int A, int B)
{
    // Base cases
    if (A == 0)
        return 0;
    if (B == 0)
        return 1;
  
    // If B is even
    long y;
    if (B % 2 == 0) 
    {
        y = exponentMod(A, B / 2);
        y = (y * y) % mod;
    }
  
    // If B is odd
    else 
    {
        y = A % mod;
        y = (y * exponentMod(A, B - 1) % 
                                  mod) % mod;
    }
  
    return (int)((y + mod) % mod);
}
  
// Function to count the number of
// ways to form the group having
// one leader
static void countWays(int N)
{
  
    // Find 2^(N-1) using modular
    // exponentiation
    long select = exponentMod(2, N - 1);
  
    // Count total ways
    long ways = ((N % mod) * (select % mod));
  
    ways %= mod;
  
    // Print the total ways
    System.out.print(ways);
}
  
// Driver Code
public static void main(String[] args)
{
  
    // Given N number of peoples
    int N = 5;
  
    // Function Call
    countWays(N);
}
}
  
// This code is contributed by sapnasingh4991

Python3

# Python3 program for the above approach
mod = 1000000007
  
# Function to find 2^x using
# modular exponentiation
def exponentMod(A, B):
      
    # Base cases
    if (A == 0):
        return 0;
    if (B == 0):
        return 1;
  
    # If B is even
    y = 0;
      
    if (B % 2 == 0):
        y = exponentMod(A, B // 2);
        y = (y * y) % mod;
  
    # If B is odd
    else:
        y = A % mod;
        y = (y * exponentMod(A, B - 1) %
                                  mod) % mod;
                               
    return ((y + mod) % mod);
  
# Function to count the number of
# ways to form the group having
# one leader
def countWays(N):
      
    # Find 2^(N-1) using modular
    # exponentiation
    select = exponentMod(2, N - 1);
  
    # Count total ways
    ways = ((N % mod) * (select % mod));
  
    ways %= mod;
  
    # Print the total ways
    print(ways)
      
# Driver code        
if __name__=='__main__':
      
    # Given N number of people
    N = 5;
  
    # Function call
    countWays(N);
  
# This code is contributed by rutvik_56

C#

// C# program for the above approach
using System;
class GFG{
   
static long mod = 1000000007;
   
// Function to find 2^x using
// modular exponentiation
static int exponentMod(int A, int B)
{
    // Base cases
    if (A == 0)
        return 0;
    if (B == 0)
        return 1;
   
    // If B is even
    long y;
    if (B % 2 == 0) 
    {
        y = exponentMod(A, B / 2);
        y = (y * y) % mod;
    }
   
    // If B is odd
    else
    {
        y = A % mod;
        y = (y * exponentMod(A, B - 1) % 
                                  mod) % mod;
    }
   
    return (int)((y + mod) % mod);
}
   
// Function to count the number of
// ways to form the group having
// one leader
static void countWays(int N)
{
   
    // Find 2^(N-1) using modular
    // exponentiation
    long select = exponentMod(2, N - 1);
   
    // Count total ways
    long ways = ((N % mod) * (select % mod));
   
    ways %= mod;
   
    // Print the total ways
    Console.Write(ways);
}
   
// Driver Code
public static void Main(String[] args)
{
   
    // Given N number of peoples
    int N = 5;
   
    // Function Call
    countWays(N);
}
}
  
// This code is contributed by sapnasingh4991

输出：

80

时间复杂度： O(log N)
辅助空间： O(1)