<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:
    using ll = long long;
    static const int N = 55;
    static const int M = 105;
    ll dp[N][M][N];
    ll pre[N][M][N];
    // pre[i][j][k] = dp[i][1][k] + dp[i][2][k] + ... + dp[i][j-1][k] + dp[i][j][k]
    int numOfArrays(int n, int m, int k) {
        const int MOD = 1e9 + 7;
        memset(dp, 0, sizeof(dp));
        memset(pre, 0, sizeof(pre));
        for(int elem = 1; elem <= m; elem++) {
            dp[1][elem][1] = 1;
            pre[1][elem][1] = elem;
        }
        for(int len = 2; len <= n; len++) {
            for(int max_elem = 1; max_elem <= m; max_elem++) {
                for(int cost = 1; cost <= k; cost++) {
                    // if i want to keep cost
                    dp[len][max_elem][cost] = max_elem * 1LL * dp[len-1][max_elem][cost];
                    dp[len][max_elem][cost] %= MOD;
                    dp[len][max_elem][cost] += pre[len-1][max_elem-1][cost-1];
                    if(dp[len][max_elem][cost] >= MOD) dp[len][max_elem][cost] -= MOD;
                    pre[len][max_elem][cost] = dp[len][max_elem][cost] + pre[len][max_elem-1][cost];
                    if(pre[len][max_elem][cost] >= MOD) pre[len][max_elem][cost] -= MOD;
                }
            }
        }
        return pre[n][m][k];
    }
};
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