Combination Sum Calculator

Combination Sum Calculator: Find All Possible Number Combinations

What Is This Calculator?

This calculator helps you find all possible combinations of numbers from a given set that add up to a specific target value. It’s useful for:

• Solving math problems
• Financial planning
• Game strategy development
• Coding interview preparation
• Educational purposes

How It Works (The Formula)

The calculator uses a backtracking algorithm to efficiently find combinations:

1. Input: Set of numbers and target sum
2. Process:
   • Sort numbers (for efficiency)
   • Recursively try combinations
   • Backtrack when sum exceeds target
3. Output: All valid combinations

Mathematically, it finds all subsets S where:
∑ S = target (S ⊆ input numbers)

How to Use the Calculator

1. Enter your numbers (comma separated, like “2, 3, 5”)
2. Set your target sum (the total you want to reach)
3. Choose options:
   • Allow number reuse (like using “2” multiple times)
   • Treat [2,3] and [3,2] as same combination
   • Limit number of results shown
4. Click “Calculate” to see all valid combinations
5. View the chart showing combination length distribution

Key Features

✔ Fast calculation with optimized algorithm
✔ Visual results with interactive chart
✔ Flexible options for different needs
✔ Mobile-friendly design
✔ No login required - use instantly

Frequently Asked Questions (FAQs)

Q: Why can’t I find combinations for my numbers?
A: Some number sets have no valid combinations. Try adjusting your target or allowing number reuse.

Q: How large can my number set be?
A: For best performance, keep under 20 numbers. Large sets may slow calculations.

Q: Are decimal numbers supported?
A: Currently only whole numbers are supported for precise combinations.

Q: Can I save my results?
A: Yes! Use your browser’s print function (Ctrl+P) to save as PDF.

Terminology Explained

• Target Sum: The total amount your combinations should equal
• Number Reuse: Using the same number multiple times in a combination
• Unique Order: Whether [2,3] and [3,2] count as different combinations
• Combination Length: How many numbers are in each solution

Technical Notes

Algorithm Source:
Based on standard backtracking algorithms used in computer science, similar to the “subset sum problem” solution.

Performance:

• Time complexity: O(2^n) in worst case
• Optimized with early termination
• Debounced input for smooth UX

Limitations:

• Works best with positive integers
• Very large targets (>1000) may cause delays
• Browser memory limits may affect huge result sets

Important Tips

1. Start with smaller numbers to test
2. Use the “unique order” option to reduce duplicate results
3. Limit results if you expect many combinations
4. For financial uses, round amounts to whole dollars first
5. The chart helps identify most common combination lengths

Example Use Cases

• Math Homework: “Find all combinations of [1,2,3] that sum to 4”
• Budgeting: “Which expenses combine to $500?”
• Game Design: “Balance item costs that combine to unlock rewards”
• Coding Practice: “Study the algorithm for technical interviews”