<p>You have a pointer at index <code>0</code> in an array of size <code>arrLen</code>. At each step, you can move 1 position to the left, 1 position to the right in the array, or stay in the same place (The pointer should not be placed outside the array at any time).</p>

<p>Given two integers <code>steps</code> and <code>arrLen</code>, return the number of ways such that your pointer is still at index <code>0</code> after <strong>exactly</strong> <code>steps</code> steps. Since the answer may be too 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>

<pre>
<strong>Input:</strong> steps = 3, arrLen = 2
<strong>Output:</strong> 4
<strong>Explanation: </strong>There are 4 differents ways to stay at index 0 after 3 steps.
Right, Left, Stay
Stay, Right, Left
Right, Stay, Left
Stay, Stay, Stay
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> steps = 2, arrLen = 4
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are 2 differents ways to stay at index 0 after 2 steps
Right, Left
Stay, Stay
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> steps = 4, arrLen = 2
<strong>Output:</strong> 8
</pre>

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

<ul>
	<li><code>1 &lt;= steps &lt;= 500</code></li>
	<li><code>1 &lt;= arrLen &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution {
    int mod=(int)(1e9+7);
    long long solve(long long i ,long long steps,long long arrlen,vector<vector<long long>>&dp){
          if(i<0 || i>=arrlen) return 0;

          if(steps==0){
            if(i==0) return 1;
            return 0;
          }

          if(dp[i][steps]!=-1) return dp[i][steps];

          //3 options
          long long stay=solve(i,steps-1,arrlen,dp);
          long long left=solve(i-1,steps-1,arrlen,dp);
          long long right=solve(i+1,steps-1,arrlen,dp);

          return dp[i][steps]=((stay)%mod+(left)%mod+(right)%mod)%mod;
    }
public:
    int numWays(int steps, int arrLen) {
        // vector<vector<long long>>dp(arrLen+1,vector<long long>(steps+1,-1));
        // return solve(0,steps,arrLen,dp);
        //above gives a memory limit exceed
        

        //note that i jo hai woh kitna aage badh sakta hai max jinta steps ho isliye 
        //dp array mein instead of arrlen+1 
        //we can use steps+1

        vector<vector<long long>>dp(steps+1,vector<long long>(steps+1,-1));
        return solve(0,steps,arrLen,dp);
    }
};
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