<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 itertools import permutations

class Solution:
    def __init__(self):
        # Precompute factorials from 0 to 10
        self.factorial_arr = [factorial(i) for i in range(11)]

    def generate_palindromes(self, n):
        palindromes = set()
        if n == 0:
            return palindromes
        
        half_len = (n + 1) // 2
        start_num = 10 ** (half_len - 1)
        end_num = 10 ** half_len - 1
        
        if n == 1:
            start_num = 0
        
        for first_half in range(start_num, end_num + 1):
            half_str = str(first_half)
            if n % 2 == 0:
                full_palindrome = half_str + half_str[::-1]
            else:
                full_palindrome = half_str + half_str[-2::-1]
            
            if len(full_palindrome) == n:
                palindromes.add(int(full_palindrome))
        return palindromes

    def count_digits(self, number, n):
        digit_freq = [0] * 10
        num_str = str(number).zfill(n)
        
        for digit in num_str:
            digit_freq[int(digit)] += 1
        return digit_freq

    def calculate_permutations(self, digit_freq, n):
        permutation_count = 0
        for digit in range(1, 10):
            if digit_freq[digit] == 0:
                continue
            
            adjusted_freq = digit_freq[:]
            adjusted_freq[digit] -= 1
            if any(freq < 0 for freq in adjusted_freq):
                continue
            
            valid_permutations = self.factorial_arr[n - 1]
            for freq in adjusted_freq:
                if freq > 0:
                    valid_permutations //= self.factorial_arr[freq]
            permutation_count += valid_permutations
        return permutation_count

    def countGoodIntegers(self, n: int, k: int) -> int:
        palindrome_list = self.generate_palindromes(n)
        valid_palindromes = [p for p in palindrome_list if p % k == 0]
        
        unique_digit_freq = set()
        for valid_palindrome in valid_palindromes:
            digit_freq = tuple(self.count_digits(valid_palindrome, n))
            unique_digit_freq.add(digit_freq)
        
        total_good_integers = 0
        for digit_freq in unique_digit_freq:
            total_good_integers += self.calculate_permutations(list(digit_freq), n)
        
        return total_good_integers


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