Counts paths from a point to reach Origin

You are standing on a point (n, m) and you want to go to origin (0, 0) by taking steps either left or down i.e. from each point you are allowed to move either in (n-1, m) or (n, m-1). Find the number of paths from point to origin.

Examples:

Input : 3 6
Output : Number of Paths 84

Input : 3 0
Output : Number of Paths 1

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

As we are restricted to move down and left only we would run a recursive loop for each of the combinations of the
steps that can be taken.

// Recursive function to count number of paths
countPaths(n, m)
{
    // If we reach bottom or top left, we are
    // have only one way to reach (0, 0)
    if (n==0 || m==0)
        return 1;

    // Else count sum of both ways
    return (countPaths(n-1, m) + countPaths(n, m-1));
}

Below is the implementation of above steps.

C++

// CPP program to count total number of 
// paths from a point to origin 
#include<bits/stdc++.h> 
using namespace std; 
  
// Recursive function to count number of paths 
int countPaths(int n, int m) 
{ 
    // If we reach bottom or top left, we are 
    // have only one way to reach (0, 0) 
    if (n==0 || m==0) 
        return 1; 
  
    // Else count sum of both ways 
    return (countPaths(n-1, m) + countPaths(n, m-1)); 
} 
  
// Driver Code 
int main() 
{ 
    int n = 3, m = 2; 
    cout << " Number of Paths " << countPaths(n, m); 
    return 0; 
} 

Java

// Java program to count total number of 
// paths from a point to origin 
import java.io.*; 
  
class GFG { 
      
    // Recursive function to count number of paths 
    static int countPaths(int n, int m) 
    { 
        // If we reach bottom or top left, we are 
        // have only one way to reach (0, 0) 
        if (n == 0 || m == 0) 
            return 1; 
      
        // Else count sum of both ways 
        return (countPaths(n - 1, m) + countPaths(n, m - 1)); 
    } 
      
    // Driver Code 
    public static void main (String[] args) 
    { 
        int n = 3, m = 2; 
        System.out.println (" Number of Paths "
                            + countPaths(n, m)); 
          
    } 
} 
  
// This code is contributed by vt_m 

Python3

# Python program to count 
# total number of 
# paths from a point to origin 
# Recursive function to  
# count number of paths 
def countPaths(n,m): 
  
    # If we reach bottom 
    # or top left, we are 
    # have only one way to reach (0, 0) 
    if (n==0 or m==0): 
        return 1
   
    # Else count sum of both ways 
    return (countPaths(n-1, m) + countPaths(n, m-1)) 
  
# Driver Code 
n = 3
m = 2
print(" Number of Paths ", countPaths(n, m)) 
  
# This code is contributed 
# by Azkia Anam. 

C#

// C# program to count total number of 
// paths from a point to origin 
using System; 
          
public class GFG { 
      
    // Recursive function to count number 
    // of paths 
    static int countPaths(int n, int m) 
    { 
          
        // If we reach bottom or top left, 
        // we are have only one way to 
        // reach (0, 0) 
        if (n == 0 || m == 0) 
            return 1; 
      
        // Else count sum of both ways 
        return (countPaths(n - 1, m)  
                 + countPaths(n, m - 1)); 
    } 
      
    // Driver Code 
    public static void Main () 
    { 
        int n = 3, m = 2; 
          
        Console.WriteLine (" Number of"
         + " Paths " + countPaths(n, m)); 
          
    } 
} 
  
// This code is contributed by Sam007. 

PHP

<?php 
// PHP program to count total number 
// of paths from a point to origin 
  
// Recursive function to  
// count number of paths 
function countPaths($n, $m) 
{ 
      
    // If we reach bottom or  
    // top left, we are 
    // have only one way to 
    // reach (0, 0) 
    if ($n == 0 || $m == 0) 
        return 1; 
  
    // Else count sum of both ways 
    return (countPaths($n - 1, $m) +  
            countPaths($n, $m - 1)); 
} 
  
    // Driver Code 
    $n = 3;  
    $m = 2; 
    echo " Number of Paths " 
      , countPaths($n, $m); 
  
// This code is contributed by aj_36 
?> 

Output:

Number of Paths 10

We can use Dynamic Programming as there are overlapping subproblems. We can draw recursion tree to see overlapping problems. For example, in case of countPaths(4, 4), we compute countPaths(3, 3) multiple times.

C++

// CPP program to count total number of 
// paths from a point to origin 
#include<bits/stdc++.h> 
using namespace std; 
  
// DP based function to count number of paths 
int countPaths(int n, int m) 
{ 
    int dp[n+1][m+1]; 
  
    // Fill entries in bottommost row and leftmost 
    // columns 
    for (int i=0; i<=n; i++) 
      dp[i][0] = 1; 
    for (int i=0; i<=m; i++) 
      dp[0][i] = 1; 
  
    // Fill DP in bottom up manner 
    for (int i=1; i<=n; i++) 
       for (int j=1; j<=m; j++) 
          dp[i][j] = dp[i-1][j] + dp[i][j-1]; 
  
    return dp[n][m]; 
} 
  
// Driver Code 
int main() 
{ 
    int n = 3, m = 2; 
    cout << " Number of Paths " << countPaths(n, m); 
    return 0; 
} 

Java

// Java program to count total number of 
// paths from a point to origin 
import java.io.*; 
  
class GFG { 
      
    // DP based function to count number of paths 
    static int countPaths(int n, int m) 
    { 
        int dp[][] = new int[n + 1][m + 1]; 
      
        // Fill entries in bottommost row and leftmost 
        // columns 
        for (int i = 0; i <= n; i++) 
            dp[i][0] = 1; 
        for (int i = 0; i <= m; i++) 
            dp[0][i] = 1; 
      
        // Fill DP in bottom up manner 
        for (int i = 1; i <= n; i++) 
        for (int j = 1; j <= m; j++) 
            dp[i][j] = dp[i - 1][j] + dp[i][j - 1]; 
      
        return dp[n][m]; 
    } 
      
    // Driver Code 
    public static void main (String[] args) { 
        int n = 3, m = 2; 
        System.out.println(" Number of Paths "
                           + countPaths(n, m)); 
          
    } 
} 
  
// This code is contributed by vt_m 

Python 3

# Python3 program to count total  
# number of paths from a po to origin 
  
# Recursive function to count 
# number of paths 
def countPaths(n, m): 
  
    # If we reach bottom or top  
    # left, we are have only one  
    # way to reach (0, 0) 
    if (n == 0 or m == 0): 
        return 1
  
    # Else count sum of both ways 
    return (countPaths(n - 1, m) + 
            countPaths(n, m - 1)) 
  
# Driver Code 
n = 3
m = 2
print("Number of Paths", 
       countPaths(n, m)) 
  
# This code is contributed by ash264 

C#

// C# program to count total number of 
// paths from a point to origin 
using System; 
          
public class GFG { 
      
    // DP based function to count number 
    // of paths 
    static int countPaths(int n, int m) 
    { 
        int [,]dp = new int[n + 1,m + 1]; 
      
        // Fill entries in bottommost row 
        // and leftmost columns 
        for (int i = 0; i <= n; i++) 
            dp[i,0] = 1; 
        for (int i = 0; i <= m; i++) 
            dp[0,i] = 1; 
      
        // Fill DP in bottom up manner 
        for (int i = 1; i <= n; i++) 
            for (int j = 1; j <= m; j++) 
                dp[i,j] = dp[i - 1,j] 
                         + dp[i,j - 1]; 
          
        return dp[n,m]; 
    } 
      
    // Driver Code 
    public static void Main () 
    { 
        int n = 3, m = 2; 
          
        Console.WriteLine(" Number of"
        + " Paths " + countPaths(n, m)); 
          
    } 
} 
  
// This code is contributed by Sam007. 

PHP

<?php 
// PHP program to count total number of 
// paths from a point to origin 
  
// DP based function to  
// count number of paths 
function countPaths($n, $m) 
{ 
      
    //$dp[$n+1][$m+1]; 
    // Fill entries in bottommost 
    // row and leftmost columns 
    for ($i = 0; $i <= $n; $i++) 
        $dp[$i][0] = 1; 
          
    for ($i = 0; $i <= $m; $i++) 
        $dp[0][$i] = 1; 
  
    // Fill DP in bottom up manner 
    for ($i = 1; $i <= $n; $i++) 
    for ($j = 1; $j <= $m; $j++) 
        $dp[$i][$j] = $dp[$i - 1][$j] +  
                      $dp[$i][$j - 1]; 
  
    return $dp[$n][$m]; 
} 
  
    // Driver Code 
    $n = 3; 
    $m = 2; 
    echo " Number of Paths " , countPaths($n, $m); 
  
// This code is contributed by m_kit 
?> 

Output:

Number of Paths 10

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