<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 __init__(self):
        self.fac = [1] * 11
        for i in range(1, 11):
            self.fac[i] = self.fac[i - 1] * i

    def countGoodIntegers(self, n: int, k: int) -> int:
        def cnt_ways(cnts, n, tot = 0):
            for d in range(1, 10):
                if cnts[d] != 0:
                    new_cnts = cnts[:]
                    new_cnts[d] -= 1
                    if any(c < 0 for c in new_cnts):
                        continue
                    arr = self.fac[n - 1]
                    for i in range(10):
                        if new_cnts[i] > 0:
                            arr //= self.fac[new_cnts[i]]
                    tot += arr
            return tot

        def gen_pal_cnt(n, palins = [], num_cnts = set()):
            if n == 0:
                return palins, num_cnts
            hl = (n + 1) // 2
            start = 10 ** (hl - 1) if n != 0 else 0
            end = 10 ** hl - 1
            for fh in range(start, end + 1):
                sh = str(fh)
                rh = sh[::-1] if n % 2 == 0 else sh[:-1][::-1]
                sp = sh + rh
                if len(sp) == n and int(sp) % k == 0:
                    palins.append(int(sp))
                    cnts = [0] * 10
                    s = str(int(sp)).zfill(n)
                    for ch in s:
                        cnts[int(ch)] += 1
                    num_cnts.add(tuple(cnts))
            return palins, num_cnts

        palins, uniq_cnts = gen_pal_cnt(n)
        return sum(cnt_ways(list(cnts), n) for cnts in uniq_cnts)

21425