Given a setA set is a collection of distinct objects. of $n$ positive integers, find all the possible subsets of integers that sum up to a number k.

Constraints:

• $1 \leq n \leq 10$

• $1 \leq x \leq 100$, where $x$ is any member of the input set

• $1 \leq$k$\leq 10^3$

The solution combines the subsets technique and bitmasking to generate all possible subsets that sum up to k. By treating each subset as a unique combination represented by a binary number, the solution ensures that all valid subsets are considered. This makes it both comprehensive and optimized for small input sizes.

Now, let’s walk through the steps of the solution.

1. We initialize a variable, count, with the total number of possible subsets using $2^n$, where $n$ is the length of the input set.

2. We initialize a list, subsets, to store all the valid subsets that sum up to the target k.

3. We iterate through all possible subsets by exploring all possible groupings. For each number in the input set, there are two possibilities: it can be a number in a subset or be left out. This creates a situation where all possible groupings of the numbers are explored.

4. We use bitmasking to include or exclude a number in a subset. Imagine each number in the input set having a switch that can be turned on or off. If the switch is on, the subset includes the number; if it’s off, it is left out. This way, every possible combination of on and off switches represents a different subset of numbers.

5. After each subset is formed by turning on or off these switches, we calculate the sum for each subset and store it in a variable sum.

6. We continue adding elements to a subset until the sum exceeds k. If it exceeds, we immediately stop adding elements to that subset, as it can no longer meet the target, and start forming the next subset.

7. We also check whether sum of a subset matches k. If it does, we store the current subset in the subsets list.

8. Once all subsets have been evaluated, return the list subsets containing all subsets whose sum equals the target value k.

Let’s look at the following illustration to get a better understanding.