Unique Paths

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A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).

The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).

How many possible unique paths are there?

Above is a 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

1. DP.

public class Solution {    public int uniquePaths(int m, int n) {        if(m==0 || n==0) return 0;        int[] dp = new int[n];        Arrays.fill(dp, 1);        for(int i=1; i<m; i++) {            for(int j=1; j<n; j++) {                dp[j] = dp[j] + dp[j-1];            }        }        return dp[n-1];    }}

2. Combinations.

public class Solution {    public int uniquePaths(int m, int n) {        if(m==0 || n==0) return 0;        if(m==1 || n==1) return 1;                long up = 1;        long down = 1;        int big = 0;        int loop = 0;                if(m-1 >= n-1) {            big = m - 1;            loop = n - 1;        }        else {            big = n - 1;            loop = m - 1;        }        for(int j=1; j<=loop; j++) {            down *= j;        }        for(int k=big+1; k<=m+n-2; k++) {            up *= k;        }                return (int)(up/down);    }}

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