Count number of ways to reach a given score in a Matrix

Given a N x N matrix mat[][] consisting of non-negative integers, the task is to find the number of ways to reach a given score M starting from the cell (0, 0) and reaching the cell (N – 1, N – 1) by going only down(from (i, j) to (i + 1, j)) or right(from (i, j) to (i, j + 1)). Whenever a cell (i, j) is reached, total score is updated by currentScore + mat[i][j].
Examples:

Input: mat[][] = {{1, 1, 1}, 
{1, 1, 1}, 
{1, 1, 1}}, M = 4 
Output: 0 
All the paths will result in a score of 5.
Input: mat[][] = {{1, 1, 1}, 
{1, 1, 1}, 
{1, 1, 1}}, M = 5 
Output: 6 

Approach: This problem can be solved using dynamic programming. First, we need to decide the states of the DP. For every cell (i, j) and a number 0 ? X ? M, we will store the number of ways to reach that cell from the cell (0, 0) with a total score of X. Thus, our solution will use 3-dimensional dynamic programming, two for the coordinates of the cells and one for the required score value. 
The required recurrence relation will be, 

dp[i][j][M] = dp[i – 1][j][M – mat[i][j]] + dp[i][j-1][M – mat[i][j]] 

Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <iostream>
using namespace std;
#define n 3
#define MAX 30

// To store the states of dp
int dp[n][n][MAX];

// To check whether a particular state
// of dp has been solved
bool v[n][n][MAX];

// Function to find the ways
// using memoization
int findCount(int mat[][n], int i, int j, int m)
{
    // Base cases
    if (i == 0 && j == 0) {
        if (m == mat[0][0])
            return 1;
        else
            return 0;
    }

    // If required score becomes negative
    if (m < 0)
        return 0;

    if (i < 0 || j < 0)
        return 0;

    // If current state has been reached before
    if (v[i][j][m])
        return dp[i][j][m];

    // Set current state to visited
    v[i][j][m] = true;

    dp[i][j][m] = findCount(mat, i - 1, j, m - mat[i][j])
                  + findCount(mat, i, j - 1, m - mat[i][j]);
    return dp[i][j][m];
}

// Driver code
int main()
{
    int mat[n][n] = { { 1, 1, 1 },
                      { 1, 1, 1 },
                      { 1, 1, 1 } };
    int m = 5;
    cout << findCount(mat, n - 1, n - 1, m);

    return 0;
}

Java

// Java implementation of the approach
class GFG
{

static int n = 3;
static int MAX =30;

// To store the states of dp
static int dp[][][] = new int[n][n][MAX];

// To check whether a particular state
// of dp has been solved
static boolean v[][][] = new boolean[n][n][MAX];

// Function to find the ways
// using memoization
static int findCount(int mat[][], int i, int j, int m)
{
    // Base cases
    if (i == 0 && j == 0)
    {
        if (m == mat[0][0])
            return 1;
        else
            return 0;
    }

    // If required score becomes negative
    if (m < 0)
        return 0;

    if (i < 0 || j < 0)
        return 0;

    // If current state has been reached before
    if (v[i][j][m])
        return dp[i][j][m];

    // Set current state to visited
    v[i][j][m] = true;

    dp[i][j][m] = findCount(mat, i - 1, j, m - mat[i][j])
                + findCount(mat, i, j - 1, m - mat[i][j]);
    return dp[i][j][m];
}

// Driver code
public static void main(String[] args)
{
    int mat[][] = { { 1, 1, 1 },
                    { 1, 1, 1 },
                    { 1, 1, 1 } };
    int m = 5;
    System.out.println(findCount(mat, n - 1, n - 1, m));
}
}

/* This code contributed by PrinciRaj1992 */

Python3

# Python3 implementation of the approach
n = 3
MAX = 60

# To store the states of dp
dp = [[[0 for i in range(30)]
          for i in range(30)]
          for i in range(MAX + 1)]

# To check whether a particular state
# of dp has been solved
v = [[[0 for i in range(30)]
         for i in range(30)]
         for i in range(MAX + 1)]

# Function to find the ways
# using memoization
def findCount(mat, i, j, m):

    # Base cases
    if (i == 0 and j == 0):
        if (m == mat[0][0]):
            return 1
        else:
            return 0

    # If required score becomes negative
    if (m < 0):
        return 0

    if (i < 0 or j < 0):
        return 0

    # If current state has been reached before
    if (v[i][j][m] > 0):
        return dp[i][j][m]

    # Set current state to visited
    v[i][j][m] = True

    dp[i][j][m] = (findCount(mat, i - 1, j,
                             m - mat[i][j]) +
                   findCount(mat, i, j - 1,
                             m - mat[i][j]))

    return dp[i][j][m]

# Driver code
mat = [ [ 1, 1, 1 ],
        [ 1, 1, 1 ],
        [ 1, 1, 1 ] ]
m = 5
print(findCount(mat, n - 1, n - 1, m))

# This code is contributed by mohit kumar

C#

// C# implementation of the approach
using System;

class GFG
{

static int n = 3;
static int MAX = 30;

// To store the states of dp
static int [,,]dp = new int[n, n, MAX];

// To check whether a particular state
// of dp has been solved
static bool [,,]v = new bool[n, n, MAX];

// Function to find the ways
// using memoization
static int findCount(int [,]mat, int i,
                          int j, int m)
{
    // Base cases
    if (i == 0 && j == 0)
    {
        if (m == mat[0, 0])
            return 1;
        else
            return 0;
    }

    // If required score becomes negative
    if (m < 0)
        return 0;

    if (i < 0 || j < 0)
        return 0;

    // If current state has been reached before
    if (v[i, j, m])
        return dp[i, j, m];

    // Set current state to visited
    v[i, j, m] = true;

    dp[i, j, m] = findCount(mat, i - 1, j, m - mat[i, j]) +
                  findCount(mat, i, j - 1, m - mat[i, j]);
    return dp[i, j, m];
}

// Driver code
public static void Main()
{
    int [,]mat = {{ 1, 1, 1 },
                  { 1, 1, 1 },
                  { 1, 1, 1 }};
    int m = 5;
    Console.WriteLine(findCount(mat, n - 1, n - 1, m));
}
}

// This code is contributed by Ryuga

PHP

<?php
// PHP implementation of the approach
$n = 3;
$MAX = 30;

// To store the states of dp
$dp = array($n, $n, $MAX);

// To check whether a particular state
// of dp has been solved
$v = array($n, $n, $MAX);

// Function to find the ways
// using memoization
function findCount($mat, $i, $j, $m)
{
    // Base cases
    if ($i == 0 && $j == 0)
    {
        if ($m == $mat[0][0])
            return 1;
        else
            return 0;
    }

    // If required score becomes negative
    if ($m < 0)
        return 0;

    if ($i < 0 || $j < 0)
        return 0;

    // If current state has been reached before
    if ($v[$i][$j][$m])
        return $dp[$i][$j][$m];

    // Set current state to visited
    $v[$i][$j][$m] = true;

    $dp[$i][$j][$m] = findCount($mat, $i - 1, $j,
                                $m - $mat[$i][$j]) +
                      findCount($mat, $i, $j - 1,
                                $m - $mat[$i][$j]);
    return $dp[$i][$j][$m];
}

// Driver code
$mat = array(array(1, 1, 1 ),
             array(1, 1, 1 ),
             array(1, 1, 1 ));
$m = 5;
echo(findCount($mat, $n - 1, $n - 1, $m));

// This code contributed by Code_Mech

Javascript

<script>

// Javascript implementation of the approach
var n = 3;
var MAX = 30;

// To store the states of dp
var dp = Array(n);
for(var i =0; i<n; i++)
{
    dp[i] = Array(n);
    for(var j=0; j<n; j++)
    {
        dp[i][j]=Array(MAX);
    }
}

// To check whether a particular state
// of dp has been solved
var v = Array(n);
for(var i =0; i<n; i++)
{
    v[i] = Array(n);
    for(var j=0; j<n; j++)
    {
        v[i][j]=Array(MAX);
    }
}

// Function to find the ways
// using memoization
function findCount(mat, i, j, m)
{
    // Base cases
    if (i == 0 && j == 0) {
        if (m == mat[0][0])
            return 1;
        else
            return 0;
    }

    // If required score becomes negative
    if (m < 0)
        return 0;

    if (i < 0 || j < 0)
        return 0;

    // If current state has been reached before
    if (v[i][j][m])
        return dp[i][j][m];

    // Set current state to visited
    v[i][j][m] = true;

    dp[i][j][m] = findCount(mat, i - 1, j,
                            m - mat[i][j])
                  + findCount(mat, i, j - 1,
                              m - mat[i][j]);
    return dp[i][j][m];
}

// Driver code
var mat = [ [ 1, 1, 1 ],
                  [ 1, 1, 1 ],
                  [ 1, 1, 1 ] ];
var m = 5;
document.write( findCount(mat, n - 1, n - 1, m));

</script>

Output:

6

Time complexity: O(N * N * M)

Auxiliary Space: O(N * N * MAX)