<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 math import factorial
from collections import defaultdict

class Solution:
    def countGoodIntegers(self, n: int, k: int) -> int:
        import math

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

        # Function to compute the number of n-digit numbers with a given digit count
        def count_numbers(count, n, fact):
            total = 0
            for d in range(1, 10):
                if count[d] == 0:
                    continue
                # Place digit 'd' in the first position
                c = list(count)
                c[d] -= 1
                # Calculate the number of permutations for the remaining digits
                perm = fact[n - 1]
                for i in range(10):
                    perm //= math.factorial(c[i])
                total += perm
            return total

        # Function to get the digit count as a tuple
        def get_digit_count(palindrome_str):
            count = [0] * 10
            for c in palindrome_str:
                count[int(c)] += 1
            return tuple(count)

        # Set to store unique digit counts of valid palindromes
        digit_counts = set()

        # Edge case for n=1
        if n == 1:
            for d in range(1, 10):
                if d % k == 0:
                    counts = [0] * 10
                    counts[d] = 1
                    digit_counts.add(tuple(counts))
        else:
            half_len = n // 2
            is_odd = n % 2
            # Generate first half
            start = 10**(half_len - 1) if half_len > 0 else 0
            end = 10**half_len
            for first_half in range(start, end):
                first_half_str = f"{first_half:0{half_len}d}" if half_len > 0 else ""
                # Skip if leading digit is zero
                if half_len > 0 and first_half_str[0] == '0':
                    continue
                if is_odd:
                    # Iterate over all possible middle digits
                    for mid in range(0, 10):
                        palindrome_str = first_half_str + str(mid) + first_half_str[::-1]
                        # Ensure the palindrome has exactly n digits
                        if len(palindrome_str) != n:
                            continue
                        # Check divisibility by k
                        num = int(palindrome_str)
                        if num % k == 0:
                            counts = [0] * 10
                            for c in palindrome_str:
                                counts[int(c)] += 1
                            digit_counts.add(tuple(counts))
                else:
                    # Even n, no middle digit
                    palindrome_str = first_half_str + first_half_str[::-1]
                    if len(palindrome_str) != n:
                        continue
                    if palindrome_str[0] == '0':
                        continue
                    num = int(palindrome_str)
                    if num % k == 0:
                        counts = [0] * 10
                        for c in palindrome_str:
                            counts[int(c)] += 1
                        digit_counts.add(tuple(counts))

        # Calculate the total number of good integers
        total_good = 0
        for count in digit_counts:
            total_good += count_numbers(count, n, fact)

        return total_good

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