<p>You are given an array <code>nums</code> consisting of <strong>positive</strong> integers and an integer <code>k</code>.</p>

<p><strong>Partition</strong> the array into two ordered <strong>groups</strong> such that each element is in exactly <strong>one</strong> group. A partition is called great if the <strong>sum</strong> of elements of each group is greater than or equal to <code>k</code>.</p>

<p>Return <em>the number of <strong>distinct</strong> great partitions</em>. Since the answer may be too large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>Two partitions are considered distinct if some element <code>nums[i]</code> is in different groups in the two partitions.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4], k = 4
<strong>Output:</strong> 6
<strong>Explanation:</strong> The great partitions are: ([1,2,3], [4]), ([1,3], [2,4]), ([1,4], [2,3]), ([2,3], [1,4]), ([2,4], [1,3]) and ([4], [1,2,3]).
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,3,3], k = 4
<strong>Output:</strong> 0
<strong>Explanation:</strong> There are no great partitions for this array.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [6,6], k = 2
<strong>Output:</strong> 2
<strong>Explanation:</strong> We can either put nums[0] in the first partition or in the second partition.
The great partitions will be ([6], [6]) and ([6], [6]).
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length, k &lt;= 1000</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>
</ul>

# seems like a dfs + memoization type of problem
# use dynamic programming to keep track of possibilities from a certain state
# how to save states?

# misread the question, its not spread it into k groups
# it's spread it into TWO DISTINCT GROUPS
# much easier question but still difficult

# approach 1)
# keep track of index and sum of both groups and how many numbers have been used?

class Solution:
    def countPartitions(self, nums: List[int], k: int) -> int:
        M = (int(pow(10, 9) + 7))

        if sum(nums) < 2 * k:
            return 0

        total_pos = 0
        for i in range(1, len(nums)):
            total_pos = (total_pos + math.comb(len(nums), i)) % M

        memo = {}

        # i, sum_left
        def dfs(i, s):
            if (i, s) in memo:
                return memo[(i, s)]
            
            if s <= 0:
                return 0

            if i == len(nums):
                if s != k:
                    return 1

                return 0

            ans = (dfs(i + 1, s) + dfs(i + 1, s - nums[i])) % M
            memo[(i, s)] = ans
            return ans
            
        ans = dfs(0, k)
        return (total_pos - ((2 * (ans % M)) % M)) % M
        
421100