Calculate Combinations Instantly
Use this tool to calculate how many ways you can choose r items from n total items. You can also switch on repetition and optionally list every combination when the total is small.
Tip: For standard combinations like lottery odds, leave repetition unchecked.
What Is a Number Combination?
A combination answers one of the most common counting questions: “How many different groups can I make?” In combinations, order does not matter. So, the group {2, 5, 9} is the same as {9, 2, 5}.
This is different from permutations, where order matters. If you are selecting a team, committee, hand of cards, or lottery numbers, you usually need combinations.
Combination vs. Permutation (Quick Difference)
- Combination: order does not matter.
- Permutation: order matters.
- Example: choosing 3 books to carry is a combination, but assigning gold/silver/bronze medals is a permutation.
How This Calculator Works
The calculator uses exact integer math (BigInt), so even very large values are handled correctly without floating-point rounding errors.
- Enter total items n.
- Enter chosen items r.
- Optional: enable repetition if the same item can be chosen multiple times.
- Click Calculate to get the total number of combinations.
Core Formula
Without Repetition
The classic combination formula is:
C(n, r) = n! / (r!(n - r)!)
Use this when each item can be selected at most once.
With Repetition
If repetition is allowed, use:
C(n + r - 1, r)
This model is useful for “multiset” selections, like choosing scoops of ice cream from a fixed list of flavors where repeats are allowed.
Practical Examples
1) Lottery Picks
If a game asks you to pick 6 numbers from 49, your total possible combinations are C(49, 6) = 13,983,816.
2) Committee Selection
Selecting 4 people from a 12-person team gives C(12, 4) = 495 possible committees.
3) Product Bundles
If a store lets customers pick 3 items from 8 categories and categories can repeat, you use combinations with repetition: C(8 + 3 - 1, 3) = C(10, 3) = 120.
Common Mistakes to Avoid
- Using combinations when order actually matters.
- Forgetting to enable repetition when duplicates are allowed.
- Entering a value of r greater than n when repetition is not allowed.
- Rounding large outputs instead of using exact integer results.
Why This Matters
Combination counting appears in probability, data science, finance, algorithm design, and everyday decision-making. Whether you are calculating betting odds, test cases, sampling options, or planning teams, combination math helps you reason clearly about possible outcomes.
FAQ
Can I use decimals?
No. Combinations are defined for non-negative integers only.
What happens if r is 0?
The answer is always 1, because there is exactly one way to choose nothing.
Can this handle large values?
Yes. The calculator uses BigInt and displays exact whole-number results, even for very large counts.