# Number of shortest paths to reach every cell from bottom-left cell in the grid

Given two number N and M. The task is to find the number of shortest paths to reach the cell(i, j) in the grid of size N × M when the moves started from the bottom-left corner
Note: cell(i, j) represents the ith row and jth column in the grid
Below image shows some of the shortest paths to reach cell(1, 4) in 4 × 4 grid Examples :

```Input : N = 3, M = 4
Output : 1 3 6 10
1 2 3 4
1 1 1 1

Input : N = 5, M = 2
Output : 1 5
1 4
1 3
1 2
1 1 ```

Approach : An efficient approach is to compute the grid starting from the bottom-left corner.

• The number of shortest paths to reach cell(n, i) is 1, where, 1 < = i < = M
• The number of shortest paths to reach cell(i, 1) is 1, where, 1 < = i < = N
• The number of shortest paths to reach cell(i, j) are the sum the number of shortest paths of cell(i-1, j) and (i, j+1), where, 1 < = j < = M and 1 < = i < = N

Below is the implementation of the above approach :

## C++

 `// CPP program to find number of shortest paths` `#include ` `using` `namespace` `std;`   `// Function to find number of shortest paths` `void` `NumberOfShortestPaths(``int` `n, ``int` `m)` `{` `    ``int` `a[n][m];`   `    ``for` `(``int` `i = 0; i < n; i++)` `        ``memset``(a[i], 0, ``sizeof``(a[i]));`   `    ``// Compute the grid starting from` `    ``// the bottom-left corner` `    ``for` `(``int` `i = n - 1; i >= 0; i--) {` `        ``for` `(``int` `j = 0; j < m; j++) {` `            ``if` `(j == 0 or i == n - 1)` `                ``a[i][j] = 1;` `            ``else` `                ``a[i][j] = a[i][j - 1] + a[i + 1][j];` `        ``}` `    ``}`   `    ``// Print the grid` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``for` `(``int` `j = 0; j < m; j++) {` `            ``cout << a[i][j] << ``" "``;` `        ``}` `        ``cout << endl;` `    ``}` `}`   `// Driver code` `int` `main()` `{` `    ``int` `n = 5, m = 2;`   `    ``// Function call` `    ``NumberOfShortestPaths(n, m);`   `    ``return` `0;` `}`

## Java

 `// Java program to find number of shortest paths` `import` `java.io.*;` `class` `GFG` `{`   `// Function to find number of shortest paths` `static` `void` `NumberOfShortestPaths(``int` `n, ``int` `m)` `{` `    ``int` `[][]a = ``new` `int``[n][m];`   `    ``// Compute the grid starting from` `    ``// the bottom-left corner` `    ``for` `(``int` `i = n - ``1``; i >= ``0``; i--) ` `    ``{` `        ``for` `(``int` `j = ``0``; j < m; j++) ` `        ``{` `            ``if` `(j == ``0` `|| i == n - ``1``)` `                ``a[i][j] = ``1``;` `            ``else` `                ``a[i][j] = a[i][j - ``1``] + a[i + ``1``][j];` `        ``}` `    ``}`   `    ``// Print the grid` `    ``for` `(``int` `i = ``0``; i < n; i++) ` `    ``{` `        ``for` `(``int` `j = ``0``; j < m; j++) ` `        ``{` `            ``System.out.print(a[i][j] + ``" "``);` `        ``}` `        ``System.out.println();` `    ``}` `}`   `// Driver code` `public` `static` `void` `main(String[] args)` `{` `    ``int` `n = ``5``, m = ``2``;`   `    ``// Function call` `    ``NumberOfShortestPaths(n, m);` `}` `}`   `// This code is contributed by Princi Singh`

## Python3

 `# Python 3 program to find ` `# number of shortest paths`   `# Function to find number of shortest paths` `def` `NumberOfShortestPaths(n, m):` `    ``a ``=` `[[``0` `for` `i ``in` `range``(m)]` `            ``for` `j ``in` `range``(n)]`   `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `range``(m):` `            ``a[i][j] ``=` `0`   `    ``# Compute the grid starting from` `    ``# the bottom-left corner` `    ``i ``=` `n ``-` `1` `    ``while``(i >``=` `0``):` `        ``for` `j ``in` `range``(m):` `            ``if` `(j ``=``=` `0` `or` `i ``=``=` `n ``-` `1``):` `                ``a[i][j] ``=` `1` `            ``else``:` `                ``a[i][j] ``=` `a[i][j ``-` `1``] ``+` `\` `                          ``a[i ``+` `1``][j]`   `        ``i ``-``=` `1`   `    ``# Print the grid` `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `range``(m):` `            ``print``(a[i][j], end ``=` `" "``)` `        ``print``(``"\n"``, end ``=` `"")`   `# Driver code` `if` `__name__ ``=``=` `'__main__'``:` `    ``n ``=` `5` `    ``m ``=` `2`   `    ``# Function call` `    ``NumberOfShortestPaths(n, m)` `    `  `# This code is contributed by` `# Surendra_Gangwar`

## C#

 `// C# program to find number of shortest paths` `using` `System;`   `class` `GFG` `{`   `// Function to find number of shortest paths` `static` `void` `NumberOfShortestPaths(``int` `n, ``int` `m)` `{` `    ``int` `[,]a = ``new` `int``[n, m];`   `    ``// Compute the grid starting from` `    ``// the bottom-left corner` `    ``for` `(``int` `i = n - 1; i >= 0; i--) ` `    ``{` `        ``for` `(``int` `j = 0; j < m; j++) ` `        ``{` `            ``if` `(j == 0 || i == n - 1)` `                ``a[i, j] = 1;` `            ``else` `                ``a[i, j] = a[i, j - 1] + a[i + 1, j];` `        ``}` `    ``}`   `    ``// Print the grid` `    ``for` `(``int` `i = 0; i < n; i++) ` `    ``{` `        ``for` `(``int` `j = 0; j < m; j++) ` `        ``{` `            ``Console.Write(a[i, j] + ``" "``);` `        ``}` `        ``Console.Write(``"\n"``);` `    ``}` `}`   `// Driver code` `public` `static` `void` `Main(String[] args)` `{` `    ``int` `n = 5, m = 2;`   `    ``// Function call` `    ``NumberOfShortestPaths(n, m);` `}` `}`   `// This code is contributed by PrinciRaj1992`

## Javascript

 ``

Output

```1 5
1 4
1 3
1 2
1 1
```

Time complexity: O(N × M), where N is number of rows and M is number of columns of the grid.
Auxiliary Space: O(N × M), where N is number of rows and M is number of columns of the grid.

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