<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:
    foo = 0

    def countGoodIntegers(self, n: int, k: int) -> int:
        assert self.foo == 0
        self.foo += 1
        return sum(count_rearrangements(key) for key in make_k_palindromes(n, k))


def make_k_palindromes(n, k):
    half = (n - 1) // 2
    keys = set()
    for digits in itertools.product("123456789", *["0123456789" for _ in range(half)]):
        left = "".join(digits)
        right = left[::-1]
        if n % 2 == 1:
            right = right[1:]
        cand = left + right
        if int(cand) % k == 0:
            keys.add("".join(sorted(cand)))
    return keys


def count_rearrangements(key):
    counts = collections.Counter(key)
    zeros = counts["0"]
    other_counts = tuple(sorted(v for k, v in counts.items() if k != "0"))
    return cached_count(zeros, other_counts)


@cache
def cached_count(zeros, other_counts):
    digits = "0" * zeros
    for digit, count in zip("123456789", other_counts):
        digits += digit * count
    rearrangements = set()
    for perm in itertools.permutations(digits):
        if perm[0] == '0':
            continue
        rearrangements.add(perm)
    return len(rearrangements)
    
33673