<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:
    def numOfArrays(self, n: int, m: int, k: int) -> int:
        MOD = 10**9 + 7
        
        # Memoization dictionary
        memo = {}

        def solve(length, k_remaining, max_val):
            if length == n:
                return 1 if k_remaining == 0 else 0
            
            if (length, k_remaining, max_val) in memo:
                return memo[(length, k_remaining, max_val)]
            
            total = 0
            for i in range(1, m + 1):
                if i > max_val:
                    if k_remaining > 0:
                        total += solve(length + 1, k_remaining - 1, i)
                else:
                    total += solve(length + 1, k_remaining, max_val)
                total %= MOD
            
            memo[(length, k_remaining, max_val)] = total
            return total
        
        if m < k:
            return 0
        
        result = 0
        for i in range(1, m + 1):
            result += solve(1, k - 1, i)
            result %= MOD
        
        return result

# Example usage:
sol = Solution()
print(sol.numOfArrays(2, 3, 1))  # Example call to the function

50500