Count the number of ways to divide N in k groups incrementally

Difficulty Level : Hard
Last Updated : 10 Sep, 2020

Given two integers N and K, the task is to count the number of ways to divide N into K groups of positive integers such that their sum is N and the number of elements in groups follows a non-decreasing order (i.e group[i] <= group[i+1]).
Examples:

Input: N = 8, K = 4
Output: 5
Explanation:
Their are 5 groups such that their sum is 8 and the number of positive integers in each group is 4.
[1, 1, 1, 5], [1, 1, 2, 4], [1, 1, 3, 3], [1, 2, 2, 3], [2, 2, 2, 2]
Input: N = 24, K = 5
Output: 164
Explanation:
There are 164 such groups such that their sum is 24 and number of positive integers in each group is 5

Naive Approach: We can solve this problem using recursion. At each step of recursion put all the values greater than equal to the previously computed value.
Below is the implementation of the above approach:

C++

// C++ implementation to count the
// number of ways to divide N in 
// groups such that each group 
// has K number of elements
 
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
int calculate(int pos, int prev, 
                 int left, int k)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
     
    // put all possible values 
    // greater equal to prev
    for (int i = prev; i <= left; i++) {
        answer += calculate(pos + 1, i, 
                          left - i, k);
    }
    return answer;
}
 
// Function to count the number of 
// ways to divide the number N
int countWaystoDivide(int n, int k)
{
    return calculate(0, 1, n, k);
}
 
// Driver Code
int main()
{
    int N = 8;
    int K = 4;
 
    cout << countWaystoDivide(N, K);
    return 0;
}

Java

// Java implementation to count the
// number of ways to divide N in 
// groups such that each group 
// has K number of elements
import java.util.*;
class GFG{
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev, 
                     int left, int k)
{
     
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
     
    // Put all possible values 
    // greater equal to prev
    for(int i = prev; i <= left; i++) 
    {
       answer += calculate(pos + 1, i, 
                           left - i, k);
    }
    return answer;
}
 
// Function to count the number of 
// ways to divide the number N
static int countWaystoDivide(int n, int k)
{
    return calculate(0, 1, n, k);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 8;
    int K = 4;
 
    System.out.print(countWaystoDivide(N, K));
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 implementation to count the
# number of ways to divide N in
# groups such that each group
# has K number of elements
 
# Function to count the number
# of ways to divide the number N
# in groups such that each group
# has K number of elements
def calculate(pos, prev, left, k):
     
    # Base Case
    if (pos == k):
        if (left == 0):
            return 1;
        else:
            return 0;
 
    # If N is divides completely
    # into less than k groups
    if (left == 0):
        return 0;
 
    answer = 0;
 
    # Put all possible values
    # greater equal to prev
    for i in range(prev, left + 1):
        answer += calculate(pos + 1, i,
                           left - i, k);
 
    return answer;
 
# Function to count the number of
# ways to divide the number N
def countWaystoDivide(n, k):
     
    return calculate(0, 1, n, k);
 
# Driver Code
if __name__ == '__main__':
     
    N = 8;
    K = 4;
 
    print(countWaystoDivide(N, K));
 
# This code is contributed by 29AjayKumar

C#

// C# implementation to count the
// number of ways to divide N in 
// groups such that each group 
// has K number of elements
using System;
 
class GFG{
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev, 
                     int left, int k)
{
     
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
     
    // Put all possible values 
    // greater equal to prev
    for(int i = prev; i <= left; i++) 
    {
       answer += calculate(pos + 1, i, 
                           left - i, k);
    }
    return answer;
}
 
// Function to count the number of 
// ways to divide the number N
static int countWaystoDivide(int n, int k)
{
    return calculate(0, 1, n, k);
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 8;
    int K = 4;
 
    Console.Write(countWaystoDivide(N, K));
}
}
 
// This code is contributed by Rajput-Ji

Output:

Time complexity: O(N^K)
Efficient Approach: In the previous approach we can see that we are solving the subproblems repeatedly, i.e. it follows the property of Overlapping Subproblems. So we can memoize the same using DP table.
Below is the implementation of the above approach:

C++

// C++ implementation to count the
// number of ways to divide N in 
// groups such that each group 
// has K number of elements
 
#include <bits/stdc++.h>
 
using namespace std;
 
// DP Table
int dp[500][500][500];
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
int calculate(int pos, int prev, 
                int left, int k)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
    // if N is divides completely 
    // into less than k groups
    if (left == 0)
        return 0;
 
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][prev][left] != -1)
        return dp[pos][prev][left];
 
    int answer = 0;
    // put all possible values 
    // greater equal to prev
    for (int i = prev; i <= left; i++) {
        answer += calculate(pos + 1, i, 
                           left - i, k);
    }
 
    return dp[pos][prev][left] = answer;
}
 
// Function to count the number of 
// ways to divide the number N in groups
int countWaystoDivide(int n, int k)
{
    // Intialize DP Table as -1
    memset(dp, -1, sizeof(dp));
 
    return calculate(0, 1, n, k);
}
 
// Driver Code
int main()
{
    int N = 8;
    int K = 4;
 
    cout << countWaystoDivide(N, K);
    return 0;
}

Java

// Java implementation to count the
// number of ways to divide N in 
// groups such that each group 
// has K number of elements
import java.util.*;
class GFG{
  
// DP Table
static int [][][]dp = new int[500][500][500];
  
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev, 
                     int left, int k)
{
    // Base Case
    if (pos == k) 
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
   
    // if N is divides completely 
    // into less than k groups
    if (left == 0)
        return 0;
  
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][prev][left] != -1)
        return dp[pos][prev][left];
  
    int answer = 0;
   
    // put all possible values 
    // greater equal to prev
    for (int i = prev; i <= left; i++) 
    {
        answer += calculate(pos + 1, i, 
                           left - i, k);
    }
  
    return dp[pos][prev][left] = answer;
}
  
// Function to count the number of 
// ways to divide the number N in groups
static int countWaystoDivide(int n, int k)
{
    // Intialize DP Table as -1
        for (int i = 0; i < 500; i++) 
        {
            for (int j = 0; j < 500; j++)
            {
                for (int l = 0; l < 500; l++)
                    dp[i][j][l] = -1;
            }
        }
  
    return calculate(0, 1, n, k);
}
  
// Driver Code
public static void main(String[] args)
{
    int N = 8;
    int K = 4;
  
    System.out.print(countWaystoDivide(N, K));
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 implementation to count the
# number of ways to divide N in
# groups such that each group
# has K number of elements
  
# DP Table
dp = [[[0 for i in range(50)]
             for j in range(50)]
          for j in range(50)]
 
# Function to count the number
# of ways to divide the number N
# in groups such that each group
# has K number of elements
def calculate(pos, prev, left, k):
    
    # Base Case
    if (pos == k):
        if (left == 0):
            return 1;
        else:
            return 0;
  
    # if N is divides completely
    # into less than k groups
    if (left == 0):
        return 0;
  
    # If the subproblem has been
    # solved, use the value
    if (dp[pos][prev][left] != -1):
        return dp[pos][prev][left];
  
    answer = 0;
  
    # put all possible values
    # greater equal to prev
    for i in range(prev,left+1):
        answer += calculate(pos + 1, i, 
                            left - i, k);
    dp[pos][prev][left] = answer;
    return dp[pos][prev][left];
  
# Function to count the number of
# ways to divide the number N in groups
def countWaystoDivide(n, k):
   
    # Intialize DP Table as -1
    for i in range(50):
        for j in range(50):
            for l in range(50):
                dp[i][j][l] = -1;
  
    return calculate(0, 1, n, k);
  
# Driver Code
if __name__ == '__main__':
    N = 8;
    K = 4;
  
    print(countWaystoDivide(N, K));
  
# This code is contributed by Rajput-Ji

C#

// C# implementation to count the
// number of ways to divide N in 
// groups such that each group 
// has K number of elements
using System;
class GFG{
  
// DP Table
static int [,,]dp = new int[50, 50, 50];
  
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev, 
                     int left, int k)
{
    // Base Case
    if (pos == k) 
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
   
    // if N is divides completely 
    // into less than k groups
    if (left == 0)
        return 0;
  
    // If the subproblem has been
    // solved, use the value
    if (dp[pos, prev, left] != -1)
        return dp[pos, prev, left];
  
    int answer = 0;
   
    // put all possible values 
    // greater equal to prev
    for (int i = prev; i <= left; i++) 
    {
        answer += calculate(pos + 1, i, 
                           left - i, k);
    }
  
    return dp[pos, prev, left] = answer;
}
  
// Function to count the number of 
// ways to divide the number N in groups
static int countWaystoDivide(int n, int k)
{
    // Intialize DP Table as -1
        for (int i = 0; i < 50; i++) 
        {
            for (int j = 0; j < 50; j++)
            {
                for (int l = 0; l < 50; l++)
                    dp[i, j, l] = -1;
            }
        }
  
    return calculate(0, 1, n, k);
}
  
// Driver Code
public static void Main(String[] args)
{
    int N = 8;
    int K = 4;
  
    Console.Write(countWaystoDivide(N, K));
}
}
 
// This code is contributed by gauravrajput1

Output:

Time complexity : O(N² * K)

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