<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 comb
class Solution:
    def countGoodIntegers(self, n: int, k: int) -> int:
        # get list of all k-palindromes
        palindromes = []
        pal_set = set()

        def rev(num):
            return int(str(num)[::-1])

        def num_to_tuple(num):
            counter = Counter(str(num))
            return tuple(counter[str(i)] for i in range(10))
            

        m = n // 2
        if n % 2 == 0:
            for i in range(10**(m-1), 10**m):#e.g. 10000 00001 -> 99999 99999
                palin = 10**m * i + rev(i)
                if palin % k == 0:
                    pal_set.add(num_to_tuple(palin))

        else:
            for i in range(int(10**(m-1)), int(10**m)):#e.g. 1000 7 0001 -> 9999 7 9999
                for middle in range(10):
                    palin = 10**(m+1) * i + middle * (10 ** m) + rev(i)
                    if palin % k == 0:
                        pal_set.add(num_to_tuple(palin))
        
        res = 0
        
        def num_rearrange(pal_tup, n):
            ret = 1
            top = n
            for dig, freq in enumerate(pal_tup):
                ret *= comb(top, freq)
                top -= freq
            return ret

        def num_rearrangements_not_zero_start(pal_tup):
            if pal_tup[0] == 0:
                return num_rearrange(pal_tup, n)
            
            pal_tup0 = list(pal_tup)
            pal_tup0[0] -= 1
            pal_tup0 = tuple(pal_tup0)
            
            return num_rearrange(pal_tup, n) - num_rearrange(pal_tup0, n-1)

        for pal_tup in pal_set:
            res += num_rearrangements_not_zero_start(pal_tup)

        return res


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