Combination Calculator
Calculate nCr (n choose r) combinations when order does not matter.
What is a combination calculator?
A combination calculator finds how many ways you can choose r items from n items when order does not matter. That count is nCr, also written C(n,r) or “n choose r,” using C(n,r) = n! / [r!(n−r)!].
How to use this nCr calculator
- Enter n, the total number of available items.
- Enter r, the number of items to select.
- For standard combinations, keep 0 ≤ r ≤ n.
- The result updates as you type. Use Calculate to confirm, Escape to reset.
- Open Show steps for the formula and the cancelled product.
- Switch to With repetition when a type may be selected more than once.
Combination formula and methodology
Standard combinations use C(n,r) = n! / [r!(n−r)!]. n is the pool. r is the subset size. The r! in the denominator removes the r! ways to order the same group. This calculator does not build n! when it can avoid it. It multiplies k = min(r, n−r) exact terms: each (acc × (n−k+i)) / i is an integer. Arithmetic is BigInt inside the supported range (n ≤ 5,000 and that reduced k ≤ 1,500).
Combination with repetition
If the same type can be chosen more than once and order still does not matter, the count is C(n+r−1, r). Five flavors and three scoops is C(7,3) = 35. That is not a reordering of a standard combination, and it is not n^r (which would count order).
Combination vs permutation
Combination: ABC, BAC, and CBA are the same group. Permutation: those three listings are different. Use this page for committees, card hands, and lottery selections. Use the Permutation Calculator for rankings, podiums, and ordered codes.
| Compare | Combination | Permutation |
|---|---|---|
| Notation | nCr · C(n,r) · n choose r | nPr |
| Order | Does not matter | Matters |
| 10 items, choose 3 | 10C3 = 120 | 10P3 = 720 |
Real-world combination examples
- Committee: 10 people, choose 3 → 10C3 = 120. Name order on the roster does not create a new committee.
- Poker hand: 52 cards, choose 5 → 52C5 = 2,598,960. A hand is a set of cards.
- Lottery line: 49 numbers, choose 6 → 49C6 = 13,983,816. That is the number of unordered tickets, not a payout probability.
- Team: 12 players, choose 5 → 12C5 = 792.
Common combination values
| n | r | nCr |
|---|---|---|
| 5 | 2 | 10 |
| 10 | 3 | 120 |
| 10 | 5 | 252 |
| 12 | 5 | 792 |
| 52 | 5 | 2,598,960 |
| 49 | 6 | 13,983,816 |
Common combination mistakes
- Using nCr when the problem ranks or orders the selection.
- Entering r greater than n in standard mode.
- Using a permutation when the question asks for an unordered group.
- Forgetting that standard combinations do not reuse an item.
- Treating a combination count as a probability.
- Expanding huge factorials by hand instead of cancelling terms.
Combination vs probability
A combination answers “how many groups are possible?” Probability answers “how likely is an event?” After you have favorable and total counts, use the Probability Calculator. n! alone lives on the Factorial Calculator.
Frequently asked questions
What is a combination calculator?
It counts unordered selections: how many groups of r you can form from n items. The result is nCr, not a ranking and not a probability.
What does nCr mean?
nCr is “n choose r.” n is the pool size and r is how many you take. It is the same value as C(n,r) and the binomial coefficient.
What is n choose r?
n choose r is the number of ways to pick r items from n when order does not matter. 10 choose 3 is 120.
When should I use combinations instead of permutations?
Use combinations when the group is what matters—a committee, a hand, a lottery line. Use permutations when sequence or rank matters.
What is the combination formula?
C(n,r) = n! / [r!(n−r)!] for 0 ≤ r ≤ n. This calculator evaluates it as a product of min(r, n−r) exact terms.
What happens when r is greater than n?
In standard mode that is invalid: you cannot choose more distinct items than exist. In repetition mode r may exceed n.
What is 0 choose 0?
0C0 = 1. There is one empty selection from an empty pool.
Can combinations be calculated with repetition?
Yes. With repetition uses C(n+r−1, r). Five flavors and three scoops is C(7,3) = 35.
What is the difference between nCr and nPr?
nCr ignores order. nPr counts every ordering. 10C3 = 120 and 10P3 = 720 because 720 / 3! = 120.
What is 52 choose 5?
52C5 = 2,598,960. That is the number of five-card hands from a 52-card deck.
What are combinations used for?
Committees, teams, card hands, lottery lines, and any count of subsets. They are also binomial coefficients in (x+y)^n.
Is this combination calculator free?
Yes. There is no signup and no paid result wall.
Does the calculator run in the browser?
Yes. n and r stay on your device. Share links store only n, r, and mode in the URL hash.
Related calculators
- Permutation Calculator — Count ordered arrangements (nPr) when rank or sequence matters.
- Factorial Calculator — Compute n! on its own, without choosing r.
- Probability Calculator — Turn a count of outcomes into event probability.
- Prime Number Calculator — Test primes that often show up in combinatorics examples.
- Exponent Calculator — Powers and roots used in related counting formulas.
- Fraction Calculator — Simplify ratios after you have a combination count.
How results are calculated
C(n,r) = n! / [r!(n−r)!]. Repetition uses C(n+r−1, r). Limits: n ≤ 5000 and reduced k ≤ 1500. Exact BigInt product after r = min(r, n−r). No floating-point nCr. n and r are calculated in the browser. They are not uploaded. No external data feed is used. The calculator uses exact integer arithmetic within its supported input range.