Question

Need help with this for practice! Please answer swiftly or I will not upvote.

A computer programmer lives in his home located at the upper left corner of the street grid pictured here and works in a building located at the lower right corner. The pictured grid's lines represent streets. On his way to work, the programmer drives along any street and can tyen at any intersection but can only go down or right and cannot go back. How many different paths are there for the programmer to drive from home to work? Please make and describe an algorithm, define every variable, write the recurrent formulas and boundary conditions to solve this issue, and please show all necessary calculations to find the total number of paths. Do not just put it all into words or else I will downvote.

home work

Answer 1

To count all the possible paths from top left to bottom right of a m x n matrix with the constraints that from each cell we can either move only to right or down and cannot go back.

N and M, denoting the number of rows and number of column respectively. count of all the possible paths from top left to bottom right of a m x n matrix.

Input : m = 2, n = 2;

Output : 2

There are two paths

(0, 0) --> (0, 1) --> (1, 1)

(0, 0) --> (1, 0) --> (1, 1)

Once approach is to generate all paths and then, determine which paths are valid.

The keys involved are:

Java program to Print all possible paths from

top left to bottom right of a mXn matrix

Java program to Print all possible Routes from

top left to bottom right of a mXn matrix

starting of mXn matrix

   i, j: Current position (For the first call use 0,0)

   m, n: Dimentions of given the matrix

Here m = 6 and n = 4, we start from (0, 0) and go to the end (5, 3) we can consider any one path lets say we choose

(0, 0) -> (0, 1) -> (0, 2) -> (1, 2) -> (2, 2) -> (3, 2) -> (4, 2)

Therefore, we moved 3 steps to the right and 5 steps downwards. Even if we take any other path same number of right and down steps will be required.

======================================================

public class FindPath

{     private static void showRoute(int mat[][], int m, int n, int i, int j, int Route[], int idx)     {

        Route[idx] = mat[i][j];

   if (i == m - 1)         {

            for (int k = j + 1; k < n; k++)           {

                Route[idx + k - j] = mat[i][k];

            }

            for (int l = 0; l < idx + n - j; l++)             {

                System.out.print(Route[l] + " ");

            }

            System.out.println();

            return;

        // Reached the right corner of the matrix we are left with only the downward movement.

        if (j == n - 1)         {

            for (int k = i + 1; k < m; k++) {

                Route[idx + k - i] = mat[k][j];

            }

            for (int l = 0; l < idx + m - i; l++)   {

                System.out.print(Route[l] + " ");

            }

            System.out.println();

            return;

        }

        // Print all the Routes that are possible after moving down

        showRoute(mat, m, n, i + 1, j, Route, idx + 1);

         // Print all the Routes that are possible after moving right

        showRoute(mat, m, n, i, j + 1, Route, idx + 1);

    }

    // Test code

    public static void main(String[] args) {

        int m = 4; //number of rows

        int n = 6; //number of cols

        int mat[][] = { { 1, 2, 3, 4,5,6 }, { 7, 8, 9 ,10,11,12},                                                                                     {13,14,15,16,17,18},{19,20,21,22,23,24}}; //m * n matrix

        int totalLengthOfRoute = m + n - 1;

        showRoute(mat, m, n, 0, 0, new int[totalLengthOfRoute], 0);

    }

}

1	2	3	4	5	6
7	8	9	10	11	12
13	14	15	16	17	18
19	20	21	22	23	24

1 7 13 19 20 21 22 23 24
1 7 13 14 20 21 22 23 24
1 7 13 14 15 21 22 23 24
1 7 13 14 15 16 22 23 24
1 7 13 14 15 16 17 23 24
1 7 13 14 15 16 17 18 24
1 7 8 14 20 21 22 23 24
1 7 8 14 15 21 22 23 24
1 7 8 14 15 16 22 23 24
1 7 8 14 15 16 17 23 24
1 7 8 14 15 16 17 18 24
1 7 8 9 15 21 22 23 24
1 7 8 9 15 16 22 23 24
1 7 8 9 15 16 17 23 24
1 7 8 9 15 16 17 18 24
1 7 8 9 10 16 22 23 24
1 7 8 9 10 16 17 23 24
1 7 8 9 10 16 17 18 24
1 7 8 9 10 11 17 23 24
1 7 8 9 10 11 17 18 24
1 7 8 9 10 11 12 18 24
1 2 8 14 20 21 22 23 24
1 2 8 14 15 21 22 23 24
1 2 8 14 15 16 22 23 24
1 2 8 14 15 16 17 23 24
1 2 8 14 15 16 17 18 24
1 2 8 9 15 21 22 23 24
1 2 8 9 15 16 22 23 24
1 2 8 9 15 16 17 23 24
1 2 8 9 15 16 17 18 24
1 2 8 9 10 16 22 23 24
1 2 8 9 10 16 17 23 24
1 2 8 9 10 16 17 18 24
1 2 8 9 10 11 17 23 24
1 2 8 9 10 11 17 18 24
1 2 8 9 10 11 12 18 24
1 2 3 9 15 21 22 23 24
1 2 3 9 15 16 22 23 24
1 2 3 9 15 16 17 23 24
1 2 3 9 15 16 17 18 24
1 2 3 9 10 16 22 23 24
1 2 3 9 10 16 17 23 24
1 2 3 9 10 16 17 18 24
1 2 3 9 10 11 17 23 24
1 2 3 9 10 11 17 18 24
1 2 3 9 10 11 12 18 24
1 2 3 4 10 16 22 23 24
1 2 3 4 10 16 17 23 24
1 2 3 4 10 16 17 18 24
1 2 3 4 10 11 17 23 24
1 2 3 4 10 11 17 18 24
1 2 3 4 10 11 12 18 24
1 2 3 4 5 11 17 23 24
1 2 3 4 5 11 17 18 24
1 2 3 4 5 11 12 18 24
1 2 3 4 5 6 12 18 24

56