<p>You are given an <code>m x n</code> binary matrix <code>matrix</code> and an integer <code>numSelect</code>.</p>

<p>Your goal is to select exactly <code>numSelect</code> <strong>distinct </strong>columns from <code>matrix</code> such that you cover as many rows as possible.</p>

<p>A row is considered <strong>covered</strong> if all the <code>1</code>&#39;s in that row are also part of a column that you have selected. If a row does not have any <code>1</code>s, it is also considered covered.</p>

<p>More formally, let us consider <code>selected = {c<sub>1</sub>, c<sub>2</sub>, ...., c<sub>numSelect</sub>}</code> as the set of columns selected by you. A row <code>i</code> is <strong>covered</strong> by <code>selected</code> if:</p>

<ul>
	<li>For each cell where <code>matrix[i][j] == 1</code>, the column <code>j</code> is in <code>selected</code>.</li>
	<li>Or, no cell in row <code>i</code> has a value of <code>1</code>.</li>
</ul>

<p>Return the <strong>maximum</strong> number of rows that can be <strong>covered</strong> by a set of <code>numSelect</code> columns.</p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2022/07/14/rowscovered.png" style="width: 240px; height: 400px;" /></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">matrix = [[0,0,0],[1,0,1],[0,1,1],[0,0,1]], numSelect = 2</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>One possible way to cover 3 rows is shown in the diagram above.<br />
We choose s = {0, 2}.<br />
- Row 0 is covered because it has no occurrences of 1.<br />
- Row 1 is covered because the columns with value 1, i.e. 0 and 2 are present in s.<br />
- Row 2 is not covered because matrix[2][1] == 1 but 1 is not present in s.<br />
- Row 3 is covered because matrix[2][2] == 1 and 2 is present in s.<br />
Thus, we can cover three rows.<br />
Note that s = {1, 2} will also cover 3 rows, but it can be shown that no more than three rows can be covered.</p>
</div>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2022/07/14/rowscovered2.png" style="height: 250px; width: 84px;" /></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">matrix = [[1],[0]], numSelect = 1</span></p>

<p><strong>Output:</strong> <span class="example-io">2</span></p>

<p><strong>Explanation:</strong></p>

<p>Selecting the only column will result in both rows being covered since the entire matrix is selected.</p>
</div>

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

<ul>
	<li><code>m == matrix.length</code></li>
	<li><code>n == matrix[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 12</code></li>
	<li><code>matrix[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
	<li><code>1 &lt;= numSelect&nbsp;&lt;= n</code></li>
</ul>

class Solution:
    def maximumRows(self, matrix: List[List[int]], numSelect: int) -> int:
        M = len(matrix)
        N = len(matrix[0])
        res = []

        def check(selected):
            count = 0
            for row in matrix:
                valid_row = True
                for c in range(N):
                    if row[c] == 1 and c not in selected:
                        valid_row = False
                if valid_row:
                    count += 1

            return count

        def backtrack(selected, col, n, k):
            if len(selected) == k:
                res.append(check(selected))
                return

            for j in range(col+1, n):
                selected.append(j)
                backtrack(selected, j, n, k)
                selected.pop()


        backtrack([], -1, N, numSelect)
        return max(res)
            
            
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