<p>Given an integer array <code>nums</code> and an integer <code>k</code>, return <code>true</code> if it is possible to divide this array into <code>k</code> non-empty subsets whose sums are all equal.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [4,3,2,3,5,2,1], k = 4
<strong>Output:</strong> true
<strong>Explanation:</strong> It is possible to divide it into 4 subsets (5), (1, 4), (2,3), (2,3) with equal sums.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4], k = 3
<strong>Output:</strong> false
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 16</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
	<li>The frequency of each element is in the range <code>[1, 4]</code>.</li>
</ul>

class Solution:
    def canPartitionKSubsets(self, nums: List[int], k: int) -> bool:


        total = sum(nums)
        if total % k != 0:
            return False
        
        target = total // k
    

        memo = dict()

        def mark(bitmask, i):
            return bitmask | 1<<i

        def unmark(bitmask, i):
            return bitmask ^ (1<<i)
        
        def is_taken(bitmask, i):
            return bitmask & (1<<i) != 0


        def dfs(start_idx, set_count, cur_sum, taken_bitmask):

 
            if set_count == k:
                return True

            if taken_bitmask in memo:
                return memo[taken_bitmask]
            
            for i in range(start_idx, len(nums)):
                new_sum = cur_sum + nums[i]
                if new_sum > target or is_taken(taken_bitmask, i):
                    continue
                new_mask = mark(taken_bitmask, i)
                if new_sum == target:
                    memo[taken_bitmask] = dfs(0, set_count+1, 0, new_mask)
                else:
                    memo[taken_bitmask] = dfs(i+1, set_count, new_sum, new_mask)
                
                if memo[taken_bitmask]:
                    return True
            memo[taken_bitmask] = False
            return False
        
        return dfs(0, 0, 0, 0)




            




    
                    
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