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3539. Find Sum of Array Product of Magical Sequences

Description

You are given two integers, m and k, and an integer array nums.

A sequence of integers seq is called magical if:

  • seq has a size of m.
  • 0 <= seq[i] < nums.length
  • The binary representation of 2seq[0] + 2seq[1] + ... + 2seq[m - 1] has k set bits.

The array product of this sequence is defined as prod(seq) = (nums[seq[0]] * nums[seq[1]] * ... * nums[seq[m - 1]]).

Return the sum of the array products for all valid magical sequences.

Since the answer may be large, return it modulo 109 + 7.

A set bit refers to a bit in the binary representation of a number that has a value of 1.

 

Example 1:

Input: m = 5, k = 5, nums = [1,10,100,10000,1000000]

Output: 991600007

Explanation:

All permutations of [0, 1, 2, 3, 4] are magical sequences, each with an array product of 1013.

Example 2:

Input: m = 2, k = 2, nums = [5,4,3,2,1]

Output: 170

Explanation:

The magical sequences are [0, 1], [0, 2], [0, 3], [0, 4], [1, 0], [1, 2], [1, 3], [1, 4], [2, 0], [2, 1], [2, 3], [2, 4], [3, 0], [3, 1], [3, 2], [3, 4], [4, 0], [4, 1], [4, 2], and [4, 3].

Example 3:

Input: m = 1, k = 1, nums = [28]

Output: 28

Explanation:

The only magical sequence is [0].

 

Constraints:

  • 1 <= k <= m <= 30
  • 1 <= nums.length <= 50
  • 1 <= nums[i] <= 108

Solutions

Solution 1

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