结对方式的数量

考虑到一个聚会中有p个人。每个人都可以作为一个人参加舞蹈，也可以与其他任何人一起参加舞蹈。任务是找到p人参加舞蹈的各种不同方式。

例子：

Input : p = 3
Output : 4
Let the three people be P1, P2 and P3
Different ways are: {P1, P2, P3}, {{P1, P2}, P3},
{{P1, P3}, P2} and {{P2, P3}, P1}.

Input : p = 2
Output : 2
The groups are: {P1, P2} and {{P1, P2}}.

方法：想法是使用动态编程来解决此问题。有两种情况：一个人作为一个人或一对参加舞蹈。对于第一种情况，问题减少到为p-1人找到解决方案。对于第二种情况，存在p-1个选项来选择要配对的个人，并且在选择了个人进行配对之后，由于p中的两个人已经配对，因此问题减少了为p-2人找到解决方案。

因此，dp的公式为：

dp[p] = dp[p-1] + (p-1) * dp[p-2].

下面是上述方法的实现：

C++

// CPP program to find number of ways to
// pair people in party
  
#include 
using namespace std;
  
// Function to find number of ways to
// pair people in party
int findWaysToPair(int p)
{
    // To store count of number of ways.
    int dp[p + 1];
  
    dp[1] = 1;
    dp[2] = 2;
  
    // Using the recurrence defined find
    // count for different values of p.
    for (int i = 3; i <= p; i++) {
        dp[i] = dp[i - 1] + (i - 1) * dp[i - 2];
    }
  
    return dp[p];
}
  
// Driver code
int main()
{
    int p = 3;
    cout << findWaysToPair(p);
    return 0;
}

Java

// Java program to find number of ways to 
// pair people in party 
  
class GFG
{
      
// Function to find number of ways to 
// pair people in party 
static int findWaysToPair(int p) 
{ 
    // To store count of number of ways. 
    int dp[] = new int[p + 1]; 
  
    dp[1] = 1; 
    dp[2] = 2; 
  
    // Using the recurrence defined find 
    // count for different values of p. 
    for (int i = 3; i <= p; i++)
    { 
        dp[i] = dp[i - 1] + (i - 1) * dp[i - 2]; 
    } 
  
    return dp[p]; 
} 
  
// Driver code 
public static void main(String args[])
{ 
    int p = 3; 
    System.out.println(findWaysToPair(p));
} 
}
  
// This code is contributed by Arnab Kundu

Python3

# Python3 program to find number of
# ways to pair people in party
  
# Function to find number of ways 
# to pair people in party
def findWays(p):
  
    # To store count of number of ways.
    dp = [0] * (p + 1)
    dp[1] = 1
    dp[2] = 2
  
    # Using the recurrence defined find
    # count for different values of p.
    for i in range(3, p + 1):
        dp[i] = (dp[i - 1] + 
                   (i - 1) * dp[i - 2])
    return dp[p]
  
# Driver code
p = 3
print(findWays(p))
  
# This code is contributed by Shrikant13

C#

// C# program to find number of ways to 
// pair people in party 
using System;
  
class GFG
{
  
// Function to find number of ways to 
// pair people in party 
public static int findWaysToPair(int p)
{
    // To store count of number of ways. 
    int[] dp = new int[p + 1];
  
    dp[1] = 1;
    dp[2] = 2;
  
    // Using the recurrence defined find 
    // count for different values of p. 
    for (int i = 3; i <= p; i++)
    {
        dp[i] = dp[i - 1] + (i - 1) * dp[i - 2];
    }
    return dp[p];
}
  
// Driver code 
public static void Main(string[] args)
{
    int p = 3;
    Console.WriteLine(findWaysToPair(p));
}
}
  
// This code is contributed by shrikanth13

输出：

4

时间复杂度： O(p)
辅助空间： O(p)