Difficulty: Hard
Tags: LeetCode, Hard, Bit Manipulation, Array, Math, Binary Search, Combinatorics, Number Theory, leetcode-3116, O(n * 2^n * logk), O(2^n), Principle of Inclusion and Exclusion

Problem

You are given an integer array coins representing coins of different denominations and an integer k.

You have an infinite number of coins of each denomination. However, you are not allowed to combine coins of different denominations.

Return the k^th smallest amount that can be made using these coins.

Example 1:

Input: coins = [3,6,9], k = 3

Output: 9

Explanation: The given coins can make the following amounts:
Coin 3 produces multiples of 3: 3, 6, 9, 12, 15, etc.
Coin 6 produces multiples of 6: 6, 12, 18, 24, etc.
Coin 9 produces multiples of 9: 9, 18, 27, 36, etc.
All of the coins combined produce: 3, 6, 9, 12, 15, etc.

Example 2:

Input: coins = [5,2], k = 7

Output: 12

Explanation: The given coins can make the following amounts:
Coin 5 produces multiples of 5: 5, 10, 15, 20, etc.
Coin 2 produces multiples of 2: 2, 4, 6, 8, 10, 12, etc.
All of the coins combined produce: 2, 4, 5, 6, 8, 10, 12, 14, 15, etc.

Constraints:

1 <= coins.length <= 15
1 <= coins[i] <= 25
1 <= k <= 2 * 10⁹
coins contains pairwise distinct integers.