Combination Calculator (nCr)
Use this free online n choose r calculator to find how many ways you can choose r items from n total items when order does not matter.
What is a combination?
A combination counts how many ways you can select items from a larger set without caring about order. For example, choosing apples, bananas, and oranges is the same group no matter the order you list them in. This is why combinations are different from permutations.
In mathematics, this is written as nCr or C(n, r), and it is also known as the binomial coefficient. People often search for this as an n choose r calculator, combination formula calculator, or probability combination calculator.
Combination formula (n choose r)
The standard formula is:
C(n, r) = n! / (r! (n - r)!)
Where:
- n = total number of available items
- r = number of items you choose
- ! = factorial (for example, 5! = 5 × 4 × 3 × 2 × 1)
This calculator uses an efficient approach that avoids giant intermediate factorial values, so it stays fast and accurate for much larger inputs than a basic manual method.
How to use this online combination calculator
- Enter the total number of items in the n field.
- Enter how many items are selected in the r field.
- Click Calculate Combination.
- Read the exact value of C(n, r), plus digit count and scientific notation for large results.
If you get an input error, check that both values are whole numbers and that r is not greater than n.
Real-world examples
1) Card hands
How many 5-card poker hands can be dealt from a 52-card deck? Compute C(52, 5) and you get 2,598,960.
2) Committee selection
If your company has 12 employees and you need a 4-person committee, the number of distinct committees is C(12, 4) = 495.
3) Lottery odds
Many lottery systems are based on combinations. If 6 numbers are chosen from 49, total possible tickets are C(49, 6) = 13,983,816. That number directly affects your jackpot odds.
Combination vs permutation
This is one of the most common points of confusion:
- Combination: order does not matter.
- Permutation: order does matter.
For example, selecting A, B, C as a team is one combination. But arranging A-B-C, A-C-B, B-A-C, etc. are different permutations.
Common mistakes to avoid
- Using decimal values. Combination inputs should be non-negative integers.
- Entering r greater than n. You cannot choose more items than exist.
- Using permutation logic when order does not matter.
- Doing factorial calculations manually for large numbers and running into overflow.
FAQ
Can I use this as a binomial coefficient calculator?
Yes. C(n, r), nCr, and binomial coefficient all refer to the same value.
What happens when r = 0 or r = n?
In both cases the answer is always 1, because there is exactly one way to choose nothing or everything.
Does this calculator support very large combinations?
Yes. It uses exact integer arithmetic with BigInt in your browser, so results stay precise even when the value has many digits.
Final thoughts
Whether you are working on probability homework, lottery analysis, card games, committee planning, or data science, a reliable online combination calculator saves time and avoids arithmetic mistakes. Enter n and r above to instantly get an accurate n choose r result.