Combination Calculator
Calculate nCr — the number of ways to choose r items from n when order doesn't matter — with exact big-number results and formula steps.
About the Combination Calculator
This combination calculator computes C(n, r), also written nCr or 'n choose r' — the number of ways to select r items from a set of n when the order of selection does not matter. Enter n and r and the exact count appears instantly, alongside the matching permutation count so you can see how much order matters.
The formula is C(n, r) = n! / (r! × (n − r)!). The tool computes it with exact big-integer arithmetic rather than floating point, so even huge results like C(1000, 500) are digit-perfect, displayed in scientific notation when they get astronomically long, with the exact value one copy-click away.
Combinations are everywhere: lottery odds, hands of cards, choosing committee members, counting pizza-topping selections, and the coefficients of the binomial theorem. If the same choice in a different order counts as different (like a PIN code), you want permutations instead — this tool shows both.
How to Use the Combination Calculator
- 1Enter n, the total number of items available.
- 2Enter r, the number of items to choose.
- 3Read C(n, r) with the formula steps, plus P(n, r) for comparison.
- 4Copy the exact value for large results.
Frequently Asked Questions
How do I calculate combinations? A worked example
How many 3-person teams can you pick from 10 people? C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 teams. Order doesn't matter — Alice, Bob, Carol is the same team as Carol, Alice, Bob.
What is the difference between combinations and permutations?
Combinations ignore order; permutations count it. From 10 people, there are C(10, 3) = 120 possible teams but P(10, 3) = 720 ways to award gold, silver and bronze — exactly 3! = 6 times more, one for each ordering of the same three people.
What are the odds of winning a 6/49 lottery?
C(49, 6) = 13,983,816, so a single ticket has a 1 in 13,983,816 chance of matching all six numbers. That's the classic real-world combination calculation — order of the drawn balls doesn't matter.
Why does C(n, 0) equal 1?
There is exactly one way to choose nothing — the empty selection. Symmetrically, C(n, n) = 1 (take everything) and in general C(n, r) = C(n, n − r): choosing 3 items to keep from 10 is the same decision as choosing 7 to leave behind.
Can this calculator handle very large n?
Yes — it uses exact integer arithmetic up to n = 5,000. C(1000, 500) has about 300 digits and is computed exactly; results longer than 24 digits are displayed in scientific notation, and the copy button gives you every digit.