<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>

// Approach-2 (Bottom Up)
// T.C : O(NKM^2)
class Solution {
public:
    /* let ways[i][j][k] = # ways to construct an array of length i with max
     * element equal to j and a search cost of k. */
    long long ways[51][101][51];
    const int MOD = 1e9 + 7;

    int numOfArrays(int n, int m, int k) {
        /* There are our base cases. For each index 1 <= j <= m, we require
           ways[1][j][1] = 1 because the array consisting of only the element j
            has length 1, maximum element j, and it has a search cost of 1. */
        for (int j = 1; j <= m; j++) {
            ways[1][j][1] = 1;
        }

        for (int a = 1; a <= n; a++) {
            for (int b = 1; b <= m; b++) {
                for (int c = 1; c <= k; c++) {
                    long long s = 0;

                    /* In this first case, we can append any element from [1, b]
                     * to the end of the array. */
                    s = (s + b * ways[a - 1][b][c]) % MOD;

                    /* In this second case, we can append the element "b" to the
                     * end of the array. */
                    for (int x = 1; x < b; x++) {
                        s = (s + ways[a - 1][x][c - 1]) % MOD;
                    }

                    ways[a][b][c] = (ways[a][b][c] + s) % MOD;
                }
            }
        }

        long long ans = 0;
        for (int j = 1; j <= m; j++) {
            ans = (ans + ways[n][j][k]) % MOD;
        }

        return int(ans);
    }
};
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