<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 get_palindromic(n: int, including_zero: bool = True) -> List[str]:
            if n == 1:
                return list(map(str, range(10)))
            if n == 2:
                return [str(i) + str(i) for i in range(0 if including_zero else 1, 10)]
            res = []
            for p in get_palindromic(n - 2):
                for i in range(0 if including_zero else 1, 10):
                    res.append(str(i) + p + str(i))
            return res
        
        @cache
        def factorial(n: int) -> int:
            if n == 0:
                return 1
            return n * factorial(n - 1)

        palindromic = get_palindromic(n, False)
        k_palindromic = []
        for p in palindromic:
            if int(p) % k == 0:
                k_palindromic.append(p)
        
        count = 0
        seen = set()
        for p in k_palindromic:
            digit_count = [0] * 10
            for d in p:
                digit_count[int(d)] += 1
            if tuple(digit_count) in seen:
                continue
            seen.add(tuple(digit_count))
            total = factorial(n)
            for c in digit_count:
                total //= factorial(c)
            if digit_count[0] > 0:
                minus = factorial(n - 1)
                minus //= factorial(digit_count[0] - 1)
                for c in digit_count[1:]:
                    minus //= factorial(c)
                total -= minus
            count += total

        return count

25419