<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:
        def generate_palindromes(n):
            palindromes = []

            if n == 1:
                return [str(i) for i in range(1, 10)]

            if n % 2 == 0:
                half_length = n // 2
                start = 10**(half_length - 1)
                end = 10**half_length
                for i in range(start, end):
                    palindromes.append(str(i) + str(i)[::-1])
            else:
                half_length = n // 2
                start = 10**(half_length - 1)
                end = 10**half_length
                for i in range(start, end):
                    for j in range(10):
                        palindromes.append(str(i) + str(j) + str(i)[::-1])

            return palindromes
        
        factorial = [1] * 11
        for i in range(1, 11):
            factorial[i] = factorial[i-1] * i

        palindromes = generate_palindromes(n)
        total = 0
        seen = set() # Keep the sorted strings in the set to not repeat again
        # If "11222211" is a palindrome and all combinations are done
        # then if "22111122" comes then it doesn't need to be added in the answer
        for pal in palindromes:
            if int(pal) % k != 0 or "".join(sorted(pal)) in seen:
                continue
                
            seen.add("".join(sorted(pal)))
            zeros = pal.count("0") if "0" in pal else 0

            count = Counter(pal)
            den, origin_den = 1, 1
            for val, cnt in count.items():
                origin_den *= factorial[cnt]
                if val == "0":
                    cnt -= 1
                den *= factorial[cnt]
            
            if zeros:
                total += (factorial[n] // origin_den) - (factorial[n-1] // den)
            else:
                total += factorial[n] // origin_den
        
        return total
23821