<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 math import factorial
from collections import Counter
from itertools import product

class Solution:
    def countGoodIntegers(self, n: int, k: int) -> int:
        MOD = 10**9 +7
        fact = [1]*(n+1)
        for i in range(1,n+1):
            fact[i] = fact[i-1]*i
        def count_perms(digits):
            cnt = Counter(digits)
            total = fact[n]
            for v in cnt.values():
                total //= fact[v]
            if '0' in cnt:
                cnt_zero = cnt.copy()
                cnt_zero['0'] -=1
                if cnt_zero['0'] ==0:
                    del cnt_zero['0']
                perms = fact[n-1]
                for v in cnt_zero.values():
                    perms //= fact[v]
                total -= perms
            return total
        seen = set()
        total_good=0
        half = n//2
        is_odd = n%2
        if is_odd:
            if half >0:
                for first_digit in '123456789':
                    for other_digits in product('0123456789', repeat=half-1):
                        first_half = first_digit + ''.join(other_digits)
                        for mid in '0123456789':
                            pal = first_half + mid + first_half[::-1]
                            num = int(pal)
                            if num%k ==0:
                                key = tuple(sorted(pal))
                                if key not in seen:
                                    seen.add(key)
                                    total_good += count_perms(pal)
            else:
                for d in '123456789':
                    pal = d
                    num = int(pal)
                    if num%k ==0:
                        key = tuple(sorted(pal))
                        if key not in seen:
                            seen.add(key)
                            total_good += count_perms(pal)
        else:
            for first_digit in '123456789':
                for other_digits in product('0123456789', repeat=half-1):
                    first_half = first_digit + ''.join(other_digits)
                    pal = first_half + first_half[::-1]
                    num = int(pal)
                    if num%k ==0:
                        key = tuple(sorted(pal))
                        if key not in seen:
                            seen.add(key)
                            total_good += count_perms(pal)
        return total_good%MOD
18496