Combination Calculator

What Is a Combination Calculator?

A combination calculator determines how many different selections can be made from a set of items when the order of the selected items does not matter. It is commonly used in mathematics, probability, statistics and other situations where you need to count possible groups or selections.

The standard combination calculation is written as nCr and uses the formula nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number selected. This calculator can also calculate combinations with repetition when the same type of item can be selected more than once.

Why Use Combination Calculator?

YesTools Combination Calculator provides fast, exact combination calculations without requiring you to work through factorials manually.

How to Use Combination Calculator

Use the calculator to determine the number of possible selections from a given number of items.

1. Enter n — Enter the total number of items or available types.
2. Enter r — Enter the number of items to be selected.
3. Choose the combination type — Select Without repetition for standard combinations or With repetition when an item type may be selected more than once.
4. Calculate the result — Select Calculate Combinations to display the exact number of possible selections.

How to Calculate Combinations Using nCr

For combinations without repetition, the standard formula is nCr = n! / (r!(n-r)!). The exclamation mark represents a factorial, which is the product of all positive integers from the specified number down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

For example, selecting 3 items from 10 items gives 10C3 = 10! / (3! × 7!) = 120. This means there are 120 different groups of three that can be selected from ten items when the order of selection does not matter.

Combinations With Repetition

When repetition is allowed, the calculation changes because the same type of item can be selected more than once. The formula is C(n+r-1,r). For example, selecting 3 items from 10 available types with repetition allowed gives C(12,3) = 220 possible combinations.

Combinations vs Permutations

The key difference between combinations and permutations is whether order matters. A combination treats selections such as A, B and B, A as the same selection. A permutation treats them as different arrangements. Combinations are therefore appropriate when you are counting groups, while permutations are used when the order or arrangement of the selected items matters.

Combination Calculator Information

The values n and r must be whole numbers from 0 to 1,000. For combinations without repetition, r cannot be greater than n because it is not possible to select more distinct items than are available. When repetition is allowed, n must be at least 1 when r is greater than zero.

The calculator uses exact JavaScript BigInt arithmetic rather than floating-point arithmetic. This allows supported combination results to remain exact even when they contain many digits. Very large results are displayed in a horizontally scrollable result area so the complete integer can still be viewed without rounding.

There is exactly one way to select zero items, so nC0 is always 1. The calculator performs all calculations locally in your browser and does not need to send the values you enter to a server.

Frequently Asked Questions