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Calculator Answers

Combinations Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A combinations calculator is a mathematical counting utility that computes the number of unique subsets of size r that can be chosen from a larger pool of n distinct elements. In combinations, the selection order is completely ignored. The calculator evaluates combinations using the standard binomial coefficient formula and applies symmetry adjustments to optimize calculation speed and accuracy. It is widely used in algebra, probability theory, lottery designs, card games, and scheduling combinations.

Enter the total number of items n and the number chosen r, and the calculator returns C(n, r), the count of unordered selections (combinations). Uses the multiplicative form for stable calculations.

Quick Answer

Calculate nCr combinations for any pool of items. Enter the pool size and selection count to find the number of unique unordered groups.

How many distinct items you have to choose from. · e.g. 10

Number to select. Cannot exceed n. · e.g. 3

Formula

C(n, r) = n! ÷ (r! × (n − r)!)

Combinations count selections where order does not matter: choosing {A, B, C} is the same as {C, B, A}. Use permutations instead when order matters.

Combinations

C(10, 3)

120

Order does not matter

n10
r3
By symmetryC(10, 3) = C(10, 7)
Result120
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Examples

C(10, 3)

120

C(52, 5) (5-card poker hands)

2,598,960

C(20, 10)

184,756

How it works

A combinations calculator counts unordered selections. The closed-form formula uses three factorials, but the calculator uses the equivalent multiplicative form to compute the answer directly, multiplying and dividing one term at a time so very large factorials never appear in intermediate values.

Closed form · C(n, r) = n! ÷ (r! × (n − r)!)

Multiplicative · Π from i = 1 to k of (n − k + i) ÷ i, where k = min(r, n − r)

Conventions: C(n, 0) = C(n, n) = 1 · C(n, r) = C(n, n − r).

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Combination CalculatorCalculate C(n, r) for unordered selections, using the multiplicative form for stability across small and large inputs.

Frequently asked questions

Combinations are selections of items where order does not matter. Picking the set {A, B, C} from a larger group is the same combination as picking {C, B, A}: only which items appear matters, not the order. The standard C(n, r) counts how many combinations of r items can be chosen from a set of n distinct items.

C(n, r) = n! ÷ (r! × (n − r)!). The numerator is all permutations of n choose r; dividing by r! removes the orderings, since each unordered set of r items appears r! times among the permutations. The calculator uses the equivalent multiplicative form to avoid computing massive factorials directly: result = product from i = 1 to r of (n − r + i) ÷ i.

Permutations count arrangements where order matters. Combinations count selections where order does not. Choosing the gold, silver, and bronze winners is a permutation. Choosing 3 finalists is a combination. P(n, r) = r! × C(n, r), so permutations are always at least as large as combinations.

C(n, r) = C(n, n − r). Choosing r items to keep is the same number of combinations as choosing the n − r items to leave out, because each choice on one side uniquely determines the other. The calculator uses this symmetry to compute with the smaller of r and n − r, which is faster and more numerically stable.

Both equal 1. There is exactly one way to choose zero items (the empty set), and exactly one way to choose all of them. This makes the formula consistent: with r = 0, the denominator is 0! × n! = n!, and the numerator is n!, giving 1.

The multiplicative formula stays numerically reasonable up to values around C(50, 25), where the answer is around 10^14. Beyond that, JavaScript's safe-integer range (about 9 × 10^15) starts to truncate digits, and the calculator flags the result as approximate. For exact answers on very large inputs, use a big-integer math tool.