<p>You are given three integers <code>n</code>, <code>m</code> and <code>k</code>. Consider the following algorithm to find the maximum element of an array of positive integers:</p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/04/02/e.png" style="width: 424px; height: 372px;" />
<p>You should build the array arr which has the following properties:</p>

<ul>
	<li><code>arr</code> has exactly <code>n</code> integers.</li>
	<li><code>1 &lt;= arr[i] &lt;= m</code> where <code>(0 &lt;= i &lt; n)</code>.</li>
	<li>After applying the mentioned algorithm to <code>arr</code>, the value <code>search_cost</code> is equal to <code>k</code>.</li>
</ul>

<p>Return <em>the number of ways</em> to build the array <code>arr</code> under the mentioned conditions. As the answer may grow large, the answer <strong>must be</strong> computed modulo <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 2, m = 3, k = 1
<strong>Output:</strong> 6
<strong>Explanation:</strong> The possible arrays are [1, 1], [2, 1], [2, 2], [3, 1], [3, 2] [3, 3]
</pre>

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

<pre>
<strong>Input:</strong> n = 5, m = 2, k = 3
<strong>Output:</strong> 0
<strong>Explanation:</strong> There are no possible arrays that satisfy the mentioned conditions.
</pre>

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

<pre>
<strong>Input:</strong> n = 9, m = 1, k = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> The only possible array is [1, 1, 1, 1, 1, 1, 1, 1, 1]
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 50</code></li>
	<li><code>1 &lt;= m &lt;= 100</code></li>
	<li><code>0 &lt;= k &lt;= n</code></li>
</ul>

class Solution {

    int MOD = 1e9+7;
    int f(int idx, int m, int n, int search_cost, int maxSoFar, 
            int k, vector<vector<vector<int>>>&dp){
            //base case
            if(idx==n){
                if(search_cost==k){
                    return 1;
                }
                return 0;
            }

            if(dp[idx][search_cost][maxSoFar] != -1){
                return dp[idx][search_cost][maxSoFar];
            }

            int result=0;
            for(int i=1; i<=m; i++){
                if(i>maxSoFar){
                    result = ((result)%MOD + f(idx+1, m, n, search_cost+1, i, k, dp))%MOD;
                }else{
                    result = ((result)%MOD + f(idx+1, m, n, search_cost, maxSoFar, k, dp))%MOD;
                }
            }

            return dp[idx][search_cost][maxSoFar] = (result%MOD);
    }
public:
    int numOfArrays(int n, int m, int k) {

        if(m<k)return 0;
        vector<vector<vector<int>>>dp(n+1, vector<vector<int>>(m+1, vector<int>(m+1, -1)));
        //f(idx, m, n, search_cost, maxSoFar);
        return f(0, m, n, 0, 0, k, dp);
    }
};
18200