<p>You are given two <strong>positive</strong> integers <code>n</code> and <code>k</code>.</p>

<p>An integer <code>x</code> is called <strong>k-palindromic</strong> if:</p>

<ul>
	<li><code>x</code> is a <span data-keyword="palindrome-integer">palindrome</span>.</li>
	<li><code>x</code> is divisible by <code>k</code>.</li>
</ul>

<p>An integer is called <strong>good</strong> if its digits can be <em>rearranged</em> to form a <strong>k-palindromic</strong> integer. For example, for <code>k = 2</code>, 2020 can be rearranged to form the <em>k-palindromic</em> integer 2002, whereas 1010 cannot be rearranged to form a <em>k-palindromic</em> integer.</p>

<p>Return the count of <strong>good</strong> integers containing <code>n</code> digits.</p>

<p><strong>Note</strong> that <em>any</em> integer must <strong>not</strong> have leading zeros, <strong>neither</strong> before <strong>nor</strong> after rearrangement. For example, 1010 <em>cannot</em> be rearranged to form 101.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 3, k = 5</span></p>

<p><strong>Output:</strong> <span class="example-io">27</span></p>

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

<p><em>Some</em> of the good integers are:</p>

<ul>
	<li>551 because it can be rearranged to form 515.</li>
	<li>525 because it is already k-palindromic.</li>
</ul>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 1, k = 4</span></p>

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

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

<p>The two good integers are 4 and 8.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 5, k = 6</span></p>

<p><strong>Output:</strong> <span class="example-io">2468</span></p>
</div>

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

<ul>
	<li><code>1 &lt;= n &lt;= 10</code></li>
	<li><code>1 &lt;= k &lt;= 9</code></li>
</ul>

class Solution:
    def countGoodIntegers(self, n: int, k: int):
        if n == 1:
            count = 0
            for i in range(1, 10):
                if i % k == 0:
                    count += 1
            return count
        palindromes = self.generate_palindromes(n)
        palindromes_filtered = set()
        for palindrome in palindromes:
            if palindrome % k == 0:
                palindromes_filtered.add(int(''.join(sorted(str(palindrome),reverse=True))))
        result = 0
        for palindrome in palindromes_filtered:
            counter = collections.Counter(str(palindrome))
            for digit in "123456789":
                if digit in counter:
                    counter[digit] -= 1
                    permutations = math.factorial(n-1) // math.prod(math.factorial(v) for v in counter.values())
                    result += permutations
                    counter[digit] += 1
        return result

    def generate_palindromes(self, n):
        lower, upper = 10**((n-2)//2), 10**((n-2)//2 + 1)
        palindromes = set()
        for i in range(int(lower), int(upper)):
            half = str(i)
            if n % 2 == 0:
                palindrome = half + half[::-1]
                palindromes.add(int(palindrome))
            else:
                for middle in range(10):
                    middle = str(middle)
                    palindrome = half + middle + half[::-1]
                    palindromes.add(int(palindrome))
        return palindromes

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