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Author: NamVr

3851-find-sum-of-array-product-of-magical-sequences/README.md

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+<p>You are given two integers, <code>m</code> and <code>k</code>, and an integer array <code>nums</code>.</p>
+A sequence of integers <code>seq</code> is called <strong>magical</strong> if:
+
+<ul>
+	<li><code>seq</code> has a size of <code>m</code>.</li>
+	<li><code>0 &lt;= seq[i] &lt; nums.length</code></li>
+	<li>The <strong>binary representation</strong> of <code>2<sup>seq[0]</sup> + 2<sup>seq[1]</sup> + ... + 2<sup>seq[m - 1]</sup></code> has <code>k</code> <strong>set bits</strong>.</li>
+</ul>
+
+<p>The <strong>array product</strong> of this sequence is defined as <code>prod(seq) = (nums[seq[0]] * nums[seq[1]] * ... * nums[seq[m - 1]])</code>.</p>
+
+<p>Return the <strong>sum</strong> of the <strong>array products</strong> for all valid <strong>magical</strong> sequences.</p>
+
+<p>Since the answer may be large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>
+
+<p>A <strong>set bit</strong> refers to a bit in the binary representation of a number that has a value of 1.</p>
+
+<p>&nbsp;</p>
+<p><strong class="example">Example 1:</strong></p>
+
+<div class="example-block">
+<p><strong>Input:</strong> <span class="example-io">m = 5, k = 5, nums = [1,10,100,10000,1000000]</span></p>
+
+<p><strong>Output:</strong> <span class="example-io">991600007</span></p>
+
+<p><strong>Explanation:</strong></p>
+
+<p>All permutations of <code>[0, 1, 2, 3, 4]</code> are magical sequences, each with an array product of 10<sup>13</sup>.</p>
+</div>
+
+<p><strong class="example">Example 2:</strong></p>
+
+<div class="example-block">
+<p><strong>Input:</strong> <span class="example-io">m = 2, k = 2, nums = [5,4,3,2,1]</span></p>
+
+<p><strong>Output:</strong> <span class="example-io">170</span></p>
+
+<p><strong>Explanation:</strong></p>
+
+<p>The magical sequences are <code>[0, 1]</code>, <code>[0, 2]</code>, <code>[0, 3]</code>, <code>[0, 4]</code>, <code>[1, 0]</code>, <code>[1, 2]</code>, <code>[1, 3]</code>, <code>[1, 4]</code>, <code>[2, 0]</code>, <code>[2, 1]</code>, <code>[2, 3]</code>, <code>[2, 4]</code>, <code>[3, 0]</code>, <code>[3, 1]</code>, <code>[3, 2]</code>, <code>[3, 4]</code>, <code>[4, 0]</code>, <code>[4, 1]</code>, <code>[4, 2]</code>, and <code>[4, 3]</code>.</p>
+</div>
+
+<p><strong class="example">Example 3:</strong></p>
+
+<div class="example-block">
+<p><strong>Input:</strong> <span class="example-io">m = 1, k = 1, nums = [28]</span></p>
+
+<p><strong>Output:</strong> <span class="example-io">28</span></p>
+
+<p><strong>Explanation:</strong></p>
+
+<p>The only magical sequence is <code>[0]</code>.</p>
+</div>
+
+<p>&nbsp;</p>
+<p><strong>Constraints:</strong></p>
+
+<ul>
+	<li><code>1 &lt;= k &lt;= m &lt;= 30</code></li>
+	<li><code>1 &lt;= nums.length &lt;= 50</code></li>
+	<li><code>1 &lt;= nums[i] &lt;= 10<sup>8</sup></code></li>
+</ul>

3851-find-sum-of-array-product-of-magical-sequences/solution.py

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+MOD = 10**9 + 7
+from functools import lru_cache
+
+class Solution:
+    def magicalSum(self, total_count: int, target_odd: int, numbers: List[int]) -> int:
+        
+        @lru_cache(None)
+        def dfs(remaining, odd_needed, index, carry):
+            if remaining < 0 or odd_needed < 0 or remaining + carry.bit_count() < odd_needed:
+                return 0
+            if remaining == 0:
+                return 1 if odd_needed == carry.bit_count() else 0
+            if index >= len(numbers):
+                return 0
+            
+            ans = 0
+            for take in range(remaining + 1):
+                ways = math.comb(remaining, take) * pow(numbers[index], take, MOD) % MOD
+                new_carry = carry + take
+                ans += ways * dfs(remaining - take, odd_needed - (new_carry % 2), index + 1, new_carry // 2)
+                ans %= MOD
+            return ans
+        
+        return dfs(total_count, target_odd, 0, 0)