<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 countGoodIntegers(self, n: int, k: int) -> int:
        facts = [factorial(i) for i in range(11)]
        
        def count_perms(d):
            buf = (n - d['0']) * facts[n - 1]
            
            for k, v in d.items():
                buf //= facts[v]
                
            #print(d, buf)
                
            return buf
        
        answer = 0
        done = set()
        
        if n % 2:
            for values in product(range(10), repeat=n // 2 + 1):
                if values[0] == 0:
                    continue
                
                values = list(str(x) for x in values)
                s = "".join(values)
                num = int(s + s[::-1][1::])
                if num % k:
                    continue
                    
                t = (tuple(sorted(values)), values[-1])
                
                if t in done:
                    continue
                
                done.add(t)
                    
                d = defaultdict(int)
                for v in values:
                    d[v] += 2
                d[values[-1]] -= 1
                
                answer += count_perms(d)
            
        else:
            for values in product(range(10), repeat=n // 2):
                if values[0] == 0:
                    continue
                
                values = list(str(x) for x in values)
                s = "".join(values)
                num = int(s + s[::-1])
                if num % k:
                    continue
                    
                t = tuple(sorted(values))
                
                if t in done:
                    continue
                
                done.add(t)
                    
                d = defaultdict(int)
                for v in values:
                    d[v] += 2
                
                answer += count_perms(d)
                
        return answer
            
20360