<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, defaultdict

        # Precompute factorials
        factorials = [1] * (n+1)
        for i in range(1, n+1):
            factorials[i] = factorials[i-1] * i

        # Function to compute permutations without leading zeros
        def permutations(counts):
            total_perms = factorials[n]
            for cnt in counts.values():
                total_perms //= factorials[cnt]

            # Compute permutations where leading digit is zero
            if counts.get(0, 0) == 0:
                perms_with_leading_zero = 0
            else:
                counts_zero = counts.copy()
                counts_zero[0] -= 1
                perms_with_leading_zero = factorials[n-1]
                for cnt in counts_zero.values():
                    perms_with_leading_zero //= factorials[cnt]
            return total_perms - perms_with_leading_zero

        unique_counts = {}
        # Generate palindromic numbers of length n
        if n == 1:
            start = 1
            end = 9
        else:
            start = 10**( (n-1)//2 )
            end = 10**( (n+1)//2 ) - 1

        for first_half in range(start, end+1):
            first_half_str = str(first_half)
            if n % 2 == 0:
                palin_str = first_half_str + first_half_str[::-1]
            else:
                palin_str = first_half_str + first_half_str[-2::-1]
            if palin_str[0] == '0':
                continue
            palin_num = int(palin_str)
            if palin_num % k == 0:
                counts = Counter(map(int, palin_str))
                counts_tuple = tuple(sorted(counts.items()))
                unique_counts[counts_tuple] = counts

        total = 0
        for counts_tuple, counts in unique_counts.items():
            perms = permutations(counts)
            total += perms

        return total

21425