<p>There is an <code>m x n</code> grid with a ball. The ball is initially at the position <code>[startRow, startColumn]</code>. You are allowed to move the ball to one of the four adjacent cells in the grid (possibly out of the grid crossing the grid boundary). You can apply <strong>at most</strong> <code>maxMove</code> moves to the ball.</p>

<p>Given the five integers <code>m</code>, <code>n</code>, <code>maxMove</code>, <code>startRow</code>, <code>startColumn</code>, return the number of paths to move the ball out of the grid boundary. Since the answer can be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/28/out_of_boundary_paths_1.png" style="width: 500px; height: 296px;" />
<pre>
<strong>Input:</strong> m = 2, n = 2, maxMove = 2, startRow = 0, startColumn = 0
<strong>Output:</strong> 6
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/28/out_of_boundary_paths_2.png" style="width: 500px; height: 293px;" />
<pre>
<strong>Input:</strong> m = 1, n = 3, maxMove = 3, startRow = 0, startColumn = 1
<strong>Output:</strong> 12
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= m, n &lt;= 50</code></li>
	<li><code>0 &lt;= maxMove &lt;= 50</code></li>
	<li><code>0 &lt;= startRow &lt; m</code></li>
	<li><code>0 &lt;= startColumn &lt; n</code></li>
</ul>

class Solution {
    vector<vector<unordered_map<int, int>>> dp;
    int numRows;
    int numCols;
    int helper(int movesLeft, int startRow, int startColumn){
        if(dp[startRow][startColumn].contains(movesLeft)){
            return dp[startRow][startColumn][movesLeft];
        }
        int ans = 0;
        if(movesLeft == 1){
            if(startRow == 0){
                ++ans;
            }
            if(startRow == numRows-1){
                ++ans;
            }
            if(startColumn == 0){
                ++ans;
            }
            if(startColumn == numCols - 1){
                ++ans;
            }
            return ans;
        }
        if(startRow != 0){
            ans = (ans + helper(movesLeft-1, startRow-1, startColumn)) % 1000000007;
        }
        if(startRow != numRows-1){
            ans = (ans + helper(movesLeft-1, startRow+1, startColumn)) % 1000000007;
        }
        if(startColumn != 0){
            ans = (ans +  helper(movesLeft-1, startRow, startColumn-1)) % 1000000007;
        }
        if(startColumn != numCols-1){
            ans = (ans + helper(movesLeft-1, startRow, startColumn+1)) % 1000000007;
        }
        dp[startRow][startColumn][movesLeft] = ans;
        return ans;
    }
public:
    int findPaths(int m, int n, int maxMove, int startRow, int startColumn) {
        numRows = m;
        numCols = n;
        dp.resize(m, vector<unordered_map<int, int>>(n));
        int ans = 0;
        for(int i = 1; i <= maxMove; ++i){
            ans = (ans + helper(i, startRow, startColumn)) % 1000000007;
        }
        return ans;
    }
};
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