Different ways to sum n using numbers greater than or equal to m

Given two natural number n and m. The task is to find the number of ways in which the numbers that are greater than or equal to m can be added to get the sum n.

Examples:

Input : n = 3, m = 1
Output : 3
Following are three different ways
to get sum n such that each term is
greater than or equal to m
1 + 1 + 1, 1 + 2, 3 

Input : n = 2, m = 1
Output : 2
Two ways are 1 + 1 and 2



The idea is to use Dynamic Programming by define 2D matrix, say dp[][]. dp[i][j] define the number of ways to get sum i using the numbers greater than or equal to j. So dp[i][j] can be defined as:

If i < j, dp[i][j] = 0, because we cannot achieve smaller sum of i using numbers greater than or equal to j.

If i = j, dp[i][j] = 1, because there is only one way to show sum i using number i which is equal to j.

Else dp[i][j] = dp[i][j+1] + dp[i-j][j], because obtaining a sum i using numbers greater than or equal to j is equal to the sum of obtaining a sum of i using numbers greater than or equal to j+1 and obtaining the sum of i-j using numbers greater than or equal to j.

Below is the implementation of this approach:

C++

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// CPP Program to find number of ways to
// which numbers that are greater than
// given number can be added to get sum.
#include <bits/stdc++.h>
#define MAX 100
using namespace std;
  
// Return number of ways to which numbers
// that are greater than given number can
// be added to get sum.
int numberofways(int n, int m)
{
    int dp[n+2][n+2];
    memset(dp, 0, sizeof(dp));
  
    dp[0][n + 1] = 1;
  
    // Filling the table. k is for numbers
    // greater than or equal that are allowed.
    for (int k = n; k >= m; k--) {
  
        // i is for sum
        for (int i = 0; i <= n; i++) {
  
            // initializing dp[i][k] to number
            // ways to get sum using numbers
            // greater than or equal k+1
            dp[i][k] = dp[i][k + 1];
  
            // if i > k
            if (i - k >= 0)
                dp[i][k] = (dp[i][k] + dp[i - k][k]);
        }
    }
  
    return dp[n][m];
}
  
// Driver Program
int main()
{
    int n = 3, m = 1;
    cout << numberofways(n, m) << endl;
    return 0;
}

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Java

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// Java Program to find number of ways to
// which numbers that are greater than
// given number can be added to get sum.
import java.io.*;
  
class GFG {
      
    // Return number of ways to which numbers
    // that are greater than given number can
    // be added to get sum.
    static int numberofways(int n, int m)
    {
        int dp[][]=new int[n+2][n+2];
          
        dp[0][n + 1] = 1;
       
        // Filling the table. k is for numbers
        // greater than or equal that are allowed.
        for (int k = n; k >= m; k--) {
       
            // i is for sum
            for (int i = 0; i <= n; i++) {
       
                // initializing dp[i][k] to number
                // ways to get sum using numbers
                // greater than or equal k+1
                dp[i][k] = dp[i][k + 1];
       
                // if i > k
                if (i - k >= 0)
                    dp[i][k] = (dp[i][k] + dp[i - k][k]);
            }
        }
       
        return dp[n][m];
    }
       
    // Driver Program
    public static void main(String args[])
    {
        int n = 3, m = 1;
        System.out.println(numberofways(n, m));
    }
}
  
/*This code is contributed by Nikita tiwari.*/

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Python3

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# Python3 Program to find number of ways to
# which numbers that are greater than
# given number can be added to get sum.
MAX = 100
import numpy as np
  
# Return number of ways to which numbers
# that are greater than given number can
# be added to get sum.
  
def numberofways(n, m) :
          
    dp = np.zeros((n + 2, n + 2))
      
    dp[0][n + 1] = 1
  
    # Filling the table. k is for numbers
    # greater than or equal that are allowed.
    for k in range(n, m - 1, -1) :
  
        # i is for sum
        for i in range(n + 1) :
  
            # initializing dp[i][k] to number
            # ways to get sum using numbers
            # greater than or equal k+1
            dp[i][k] = dp[i][k + 1]
  
            # if i > k
            if (i - k >= 0) :
                dp[i][k] = (dp[i][k] + dp[i - k][k])
  
    return dp[n][m]
  
# Driver Code
if __name__ == "__main__" :
  
    n, m = 3, 1
    print(numberofways(n, m))
  
# This code is contributed by Ryuga

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C#

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// C# program to find number of ways to
// which numbers that are greater than
// given number can be added to get sum.
using System;
  
class GFG {
  
    // Return number of ways to which numbers
    // that are greater than given number can
    // be added to get sum.
    static int numberofways(int n, int m)
    {
        int[, ] dp = new int[n + 2, n + 2];
  
        dp[0, n + 1] = 1;
  
        // Filling the table. k is for numbers
        // greater than or equal that are allowed.
        for (int k = n; k >= m; k--) {
  
            // i is for sum
            for (int i = 0; i <= n; i++) {
  
                // initializing dp[i][k] to number
                // ways to get sum using numbers
                // greater than or equal k+1
                dp[i, k] = dp[i, k + 1];
  
                // if i > k
                if (i - k >= 0)
                    dp[i, k] = (dp[i, k] + dp[i - k, k]);
            }
        }
  
        return dp[n, m];
    }
  
    // Driver Program
    public static void Main()
    {
        int n = 3, m = 1;
        Console.WriteLine(numberofways(n, m));
    }
}
  
/*This code is contributed by vt_m.*/

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PHP

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<?php 
  
// PHP Program to find number of ways to
// which numbers that are greater than
// given number can be added to get sum.
  
$MAX = 100;
  
// Return number of ways to which numbers
// that are greater than given number can
// be added to get sum.
function numberofways($n, $m)
{
    global $MAX;
    $dp = array_fill(0, $n + 2, array_fill(0, $n+2, NULL));
  
    $dp[0][$n + 1] = 1;
  
    // Filling the table. k is for numbers
    // greater than or equal that are allowed.
    for ($k = $n; $k >= $m; $k--) 
    {
  
        // i is for sum
        for ($i = 0; $i <= $n; $i++) 
        {
  
            // initializing dp[i][k] to number
            // ways to get sum using numbers
            // greater than or equal k+1
            $dp[$i][$k] = $dp[$i][$k + 1];
  
            // if i > k
            if ($i - $k >= 0)
                $dp[$i][$k] = ($dp[$i][$k] + $dp[$i - $k][$k]);
        }
    }
  
    return $dp[$n][$m];
}
  
    // Driver Program
    $n = 3;
    $m = 1;
    echo numberofways($n, $m) ;
    return 0;
      
    // This code is contributed by ChitraNayal
?>

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Output:

3


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Improved By : AnkitRai01, chitranayal