<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) -> int:
        from math import factorial
        from collections import Counter

        def generate_palindromes(n):
            palindromes = []
            half_len = (n + 1) // 2
            start = 10 ** (half_len - 1)
            end = 10 ** half_len
            for first_half in range(start, end):
                first_half_str = str(first_half)
                if n % 2 == 0:
                    palindrome_str = first_half_str + first_half_str[::-1]
                else:
                    palindrome_str = first_half_str + first_half_str[-2::-1]
                palindrome = int(palindrome_str)
                palindromes.append(palindrome)
            return palindromes

        palindromes = generate_palindromes(n)
        counts_set = set()
        for p in palindromes:
            if p % k == 0:
                p_str = str(p)
                counts = [0] * 10
                for digit in p_str:
                    counts[int(digit)] += 1
                counts_set.add(tuple(counts))

        total = 0
        for counts_tuple in counts_set:
            counts = list(counts_tuple)
            n_digits = sum(counts)
            total_perms = factorial(n_digits)
            for c in counts:
                total_perms //= factorial(c)

            # Correct calculation of permutations starting with zero
            if counts[0] > 0:
                counts_zero = counts.copy()
                counts_zero[0] -= 1
                total_invalid = factorial(n_digits - 1)
                for c in counts_zero:
                    total_invalid //= factorial(c)
                total_valid = total_perms - total_invalid
            else:
                total_valid = total_perms

            total += total_valid
        return total

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