Generating Combinations

This challenge focuses on generating all unique combinations of a given size from a set of distinct elements. Combinations are fundamental in many areas of computer science, including probability, statistics, and algorithm design, as they represent the selection of items where the order does not matter.

Problem Description

Your task is to write a function that takes two inputs: a list of distinct elements and an integer k. The function should return a list of all possible unique combinations of size k from the given list of elements. Each combination should be represented as a list or array of elements.

Key Requirements:

• The input list contains distinct elements.
• The order of elements within a combination does not matter. For example, if the input list is [1, 2, 3] and k is 2, then [1, 2] and [2, 1] are considered the same combination. Your output should only include one representation of each unique combination.
• The order of the combinations in the output list does not matter.
• If k is greater than the number of elements in the input list, an empty list of combinations should be returned.
• If k is 0, a list containing a single empty combination should be returned.

Expected Behavior: The function should systematically explore all possible selections of k elements from the input list, ensuring no duplicates are generated and each unique combination is represented exactly once.

Edge Cases:

• k is 0.
• k is equal to the size of the input list.
• k is greater than the size of the input list.
• The input list is empty.

Examples

Example 1:

Input: elements = [1, 2, 3], k = 2
Output: [[1, 2], [1, 3], [2, 3]]
Explanation: We need to find all unique pairs from [1, 2, 3].
- Combining 1 with others: [1, 2], [1, 3]
- Combining 2 with others (avoiding duplicates): [2, 3]
- Combining 3 with others (already covered)

Example 2:

Input: elements = ["a", "b", "c", "d"], k = 3
Output: [["a", "b", "c"], ["a", "b", "d"], ["a", "c", "d"], ["b", "c", "d"]]
Explanation: We need to find all unique triplets from ["a", "b", "c", "d"].

Example 3:

Input: elements = [10, 20], k = 0
Output: [[]]
Explanation: When k is 0, there is one combination: the empty set.

Example 4:

Input: elements = [5, 6, 7], k = 4
Output: []
Explanation: It's impossible to form a combination of size 4 from a list of 3 elements.

Constraints

• The number of elements in the input list will be between 0 and 15, inclusive.
• The value of k will be between 0 and 15, inclusive.
• All elements in the input list will be distinct.
• The solution should aim for a time complexity that is efficient for the given constraints.

Notes

Consider using a recursive approach. For each element, you have two choices: either include it in the current combination or exclude it. You'll need to keep track of the current combination being built and the remaining elements to consider. Think about how to avoid generating duplicate combinations by ensuring elements are picked in a specific order or by using a mechanism to mark elements as "used."