<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:
        half = (n + 1) // 2
        odd_count = n % 2
        palindromes = set()

        def helper(i, curr):
            if i == half:
                left_half = str(curr)
                right_half = left_half[::-1]
                if odd_count == 1:
                    right_half = right_half[1:]

                val = left_half + right_half
                if int(val) % k == 0:
                    val = list(val)
                    val.sort()
                    palindromes.add(''.join(val))
                return

            for digit in range(10):
                if i == 0 and digit == 0:
                    continue

                helper(i + 1, curr * 10 + digit)
        helper(0, 0)

        factorials = [0] * (n + 1)
        for i in range(n + 1):
            factorials[i] = factorial(i)

        ans = 0
        for p in palindromes:            
            freqs = [0] * 10
            
            for d in p:
                freqs[int(d)] += 1    

            divisor = 1
            for c in freqs:
                divisor *= factorials[c]

            ans += factorials[n] // divisor

            if freqs[0] > 0:
                divisor = 1

                for i in range(10):
                    c = freqs[i] if i > 0 else freqs[i] - 1
                    divisor *= factorials[c]
                ans -= factorials[n - 1] // divisor    

        return ans    
                




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