<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 collections import defaultdict
pal_tracker = defaultdict(dict)
for i in range(10**5+1):
    a = str(i)
    b = str(i)+str(i)[::-1]
    c = str(i)+str(i)[:-1][::-1]
    tmp = [a,b,c]
    for i in tmp:
        g = len(i)
        v = int(i)
        if str(v)==i and i==i[::-1]:
            for j in range(1,10):
                if v%j==0:
                    if g not in pal_tracker[j]:
                        pal_tracker[j][g] = set()
                    pal_tracker[j][g].add(tuple(sorted(i)))

from functools import lru_cache
@lru_cache(None)
def f(n):
    return 1 if n<=1 else n*f(n-1)
        
class Solution:
    def countGoodIntegers(self, n: int, k: int) -> int:
        def process(n):
            s = str(n)
            z = s.count('0')
            r = len(s)-z
            nums = r*f(r+z-1)
            tracker = defaultdict(int)
            for i in s:
                tracker[i] += 1
            denom = 1
            for i in tracker:
                denom *= f(tracker[i])
            return nums/denom
        res = 0
        for i in pal_tracker[k][n]:
            i = ''.join(i)
            v = process(i)
            res += v
        return int(res)

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