<p>Given a rectangular pizza represented as a <code>rows x cols</code>&nbsp;matrix containing the following characters: <code>&#39;A&#39;</code> (an apple) and <code>&#39;.&#39;</code> (empty cell) and given the integer <code>k</code>. You have to cut the pizza into <code>k</code> pieces using <code>k-1</code> cuts.&nbsp;</p>

<p>For each cut you choose the direction: vertical or horizontal, then you choose a cut position at the cell boundary and cut the pizza into two pieces. If you cut the pizza vertically, give the left part of the pizza to a person. If you cut the pizza horizontally, give the upper part of the pizza to a person. Give the last piece of pizza to the last person.</p>

<p><em>Return the number of ways of cutting the pizza such that each piece contains <strong>at least</strong> one apple.&nbsp;</em>Since the answer can be a huge number, return this modulo 10^9 + 7.</p>

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

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2020/04/23/ways_to_cut_apple_1.png" style="width: 500px; height: 378px;" /></strong></p>

<pre>
<strong>Input:</strong> pizza = [&quot;A..&quot;,&quot;AAA&quot;,&quot;...&quot;], k = 3
<strong>Output:</strong> 3 
<strong>Explanation:</strong> The figure above shows the three ways to cut the pizza. Note that pieces must contain at least one apple.
</pre>

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

<pre>
<strong>Input:</strong> pizza = [&quot;A..&quot;,&quot;AA.&quot;,&quot;...&quot;], k = 3
<strong>Output:</strong> 1
</pre>

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

<pre>
<strong>Input:</strong> pizza = [&quot;A..&quot;,&quot;A..&quot;,&quot;...&quot;], k = 1
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>1 &lt;= rows, cols &lt;= 50</code></li>
	<li><code>rows ==&nbsp;pizza.length</code></li>
	<li><code>cols ==&nbsp;pizza[i].length</code></li>
	<li><code>1 &lt;= k &lt;= 10</code></li>
	<li><code>pizza</code> consists of characters <code>&#39;A&#39;</code>&nbsp;and <code>&#39;.&#39;</code> only.</li>
</ul>

from collections import defaultdict
class Solution:
    def ways(self, pizza, k) -> int:
        m = len(pizza)
        n = len(pizza[0])
        MOD = 10 ** 9 + 7
        
        apples = [[0] * (n + 1) for _ in range(m + 1)]
        cut = [[0] * (n) for _ in range(m)]

        for i in range(m - 1, -1, -1):
            count = 0
            for j in range(n - 1, -1, -1):
                if pizza[i][j] == 'A':
                    count += 1
                apples[i][j] = apples[i + 1][j] + count
                if apples[i][j] > 0:
                    cut[i][j] = 1
        for c in range(1, k):
            newcut = [[0] * n for _ in range(m)]
            for i in range(m - 1, -1, -1):
                for j in range(n - 1, -1, -1):
                    horiz = sum(cut[h][j] for h in range(i + 1, m) if apples[i][j] > apples[h][j]) % MOD
                    vert = sum(cut[i][v] for v in range(j + 1, n) if apples[i][j] > apples[i][v]) % MOD
                    newcut[i][j] = (horiz + vert) % MOD
            cut = newcut
        return cut[0][0] % MOD



                

            

        
                

 

        
        
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