<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>

from collections import Counter
import math

class Solution:
    def countGoodIntegers(self, n: int, k: int) -> int:
        def generate_palindromes(n):
            if n % 2 == 0:
                half_len = n // 2
                start = 10**(half_len - 1)
                end = 10**half_len
            else:
                half_len = n // 2 + 1
                start = 10**(half_len - 1)
                end = 10**half_len

            palindromes = set()
            for i in range(start, end):
                s = str(i)
                if n % 2 == 0:
                    palindrome = s + s[::-1]
                else:
                    palindrome = s + s[-2::-1]
                if int(palindrome) % k == 0:
                    palindromes.add(palindrome)
            return palindromes

        def count_permutations(s):
            count = math.factorial(len(s))
            for v in Counter(s).values():
                count //= math.factorial(v)
            return count

        palindromes = generate_palindromes(n)
        total_count = 0
        seen = set()
        for p in palindromes:
            # Calculate permutations of the palindrome
            total_permutations = count_permutations(p)
            p_counter = Counter(p)
            if tuple(sorted(p_counter.items())) not in seen:
                seen.add(tuple(sorted(p_counter.items())))
                # Check for '0' and adjust
                if '0' in p:
                    zero_count = p.count('0')
                    # Subtract permutations where leading zeros are involved
                    total_permutations -= count_permutations(p.replace('0', '', 1))

                total_count += total_permutations

        return total_count
29146