<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 {
// public:
//     vector<vector<vector<int>>>memo;
//     int mod=1e9+7;
//     int numOfArraysfun(int ind, int searchspace, int mx,int n,int m,int k){
//         if(ind==n){
//             return (searchspace==k);
//         }
//         if(memo[ind][searchspace][mx+1]!=-1){
//             return memo[ind][searchspace][mx+1];
//         }
//         int result=0;
//         for(int i=1; i<=m; i++){
//             if(i>mx){
//                 result=(result%mod+numOfArraysfun(ind+1,searchspace+1,i,n,m,k)%mod)%mod;
//             }else{
//                 result=(result%mod+numOfArraysfun(ind+1,searchspace,mx,n,m,k)%mod)%mod;
//             }
//         }
//         return memo[ind][searchspace][mx+1] = result%mod;
//     }
//     int numOfArrays(int n, int m, int k) {
//         memo.assign(n+1,vector<vector<int>>(k+1,vector<int>(m+2,-1)));
//         return numOfArraysfun(0,0,-1,n,m,k);
//     }
// };

// 


class Solution {
public:
    vector<vector<vector<long long>>> memo;
    int mod = 1e9 + 7;

    long long numOfArraysfun(int ind, int searchspace, int mx, int n, int m, int k) {

        if (ind == n) {
            return searchspace == k;
        }

        if (memo[ind][searchspace][mx+1] != -1) {
            return memo[ind][searchspace][mx+1];
        }

        long long result = 0;
        for (int i = 1; i <= m; i++) {
            if (i > mx) {
                
                result = (result + numOfArraysfun(ind + 1, searchspace + 1, i, n, m, k)) % mod;
            } else {
                
                result = (result + numOfArraysfun(ind + 1, searchspace, mx, n, m, k)) % mod;
            }
        }



        return memo[ind][searchspace][mx+1] = result;
    }

    int numOfArrays(int n, int m, int k) {
        // memo.assign(n + 1, vector<vector<int>>(k + 1, vector<int>(m + 1, -1)));
          memo.assign(n+1, vector<vector<long long>>(n+1, vector<long long>(m+2, -1)));
        return numOfArraysfun(0, 0, 0, n, m, k);
    }
};

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