<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
class Solution:
    def countGoodIntegers(self, n: int, k: int) -> int:
        
        odd = n % 2 == 1
        numDigitsInHalf = n//2-1 if n % 2 == 0 else n//2
        lowerBound = 10**numDigitsInHalf
        higherBound = 10**(numDigitsInHalf+1)
        
        setUnique = set()
        for i in range(lowerBound, higherBound):
            s = str(i)
            if odd:
                s += s[-2::-1]
            else:
                s += s[::-1]
            if int(s) % k == 0:
                s = tuple(sorted(list(s)))
                setUnique.add(s)
        #now we have a set of all the unique first halves of the palindrome. For each one, we can 
        numEach = []
        for s in setUnique:
            numEach.append(defaultdict(int))
            for char in s:
                numEach[-1][char] += 1
            
        totalPerms = 0
        for i in range(len(numEach)):
            #the number of ways to rearrange it = number of combinations - num where 0 is at the front
            perms = math.factorial(n)
            for key in numEach[i]:
                perms //= math.factorial(numEach[i][key])
            if '0' in numEach[i]:
                numStartWithZero = math.factorial(n-1)
                numEach[i]['0'] -= 1
                for k in numEach[i]:
                    numStartWithZero //= math.factorial(numEach[i][k])
                perms -= numStartWithZero
                numEach[i]['0'] += 1
            totalPerms += perms
        return totalPerms

                


                    
            
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