Identifying Sustainable Product Combinations

Many consumers are increasingly conscious of both environmental impact and health. This challenge asks you to determine the maximum number of products that can be selected from a list, given constraints on the total number of recyclable products and the total fat content. This is a simplified model of a real-world optimization problem faced by retailers and consumers alike.

Problem Description

You are given a list of products, each characterized by its recyclability (boolean) and fat content (a numerical value). Your task is to select a subset of these products such that:

1. The total fat content of the selected products does not exceed a given maximum fat limit.
2. The number of recyclable products in the selected subset is maximized.

You must return the maximum number of recyclable products that can be included in a valid subset.

Key Requirements:

• The solution must consider all possible subsets of products.
• The solution must adhere to the fat content limit.
• The solution must prioritize maximizing the number of recyclable products.

Expected Behavior:

The function should take a list of products (each product represented as a tuple/pair of (recyclable, fat_content)) and a maximum fat limit as input. It should return an integer representing the maximum number of recyclable products that can be selected while staying within the fat limit.

Edge Cases to Consider:

• Empty product list: Should return 0.
• Maximum fat limit of 0: Should return the number of recyclable products that have a fat content of 0.
• All products are non-recyclable: Should return 0.
• All products exceed the fat limit: Should return 0.
• Products with zero fat content.
• Large number of products (consider efficiency).

Examples

Example 1:

Input: products = [(True, 5.0), (False, 2.0), (True, 1.0), (False, 8.0), (True, 3.0)], max_fat = 7.0
Output: 2
Explanation: Selecting the products (True, 1.0) and (True, 3.0) gives a total fat content of 4.0 (<= 7.0) and includes 2 recyclable products.  No other combination yields more recyclable products within the fat limit.

Example 2:

Input: products = [(True, 5.0), (False, 2.0), (True, 1.0), (False, 8.0), (True, 3.0)], max_fat = 10.0
Output: 3
Explanation: Selecting the products (True, 1.0), (True, 3.0), and (True, 5.0) gives a total fat content of 9.0 (<= 10.0) and includes 3 recyclable products.

Example 3:

Input: products = [(False, 5.0), (False, 2.0), (False, 1.0), (False, 8.0), (False, 3.0)], max_fat = 7.0
Output: 0
Explanation:  Since all products are non-recyclable, the maximum number of recyclable products is 0, regardless of the fat limit.

Constraints

• 1 <= len(products) <= 20
• Each product is represented as a tuple (recyclable, fat_content), where recyclable is a boolean and fat_content is a floating-point number.
• 0.0 <= fat_content <= 100.0
• max_fat is a floating-point number between 0.0 and 100.0.
• Performance: The solution should complete within 1 second for the given constraints. Brute-force approaches are acceptable given the small input size.

Notes

Consider using a brute-force approach (checking all possible subsets) due to the small input size. You can generate all subsets using bit manipulation or recursion. For each subset, calculate the total fat content and the number of recyclable products. Keep track of the maximum number of recyclable products found so far while respecting the fat limit. Think about how to efficiently calculate the total fat content and recyclable count for each subset.