# Combinations Calculator – Calculate nCr

Enter the total number of objects and the sample size to calculate the possible combinations with the n choose k combinations calculator.

## Result:

### Number of Combinations C(n,r)

### Number of Permutations P(n,r)

## On this page:

## How to Calculate Combinations

Combinations are a set of items from a larger collection without regard to the order of the items in the set. Combinations are often represented as *nCr* or *n choose r*.

Lottery tickets are a common example to demonstrate what combinations are. Each ticket represents a combination of numbers from a larger set.

In fact, once you know the total number of combinations, then you can calculate the probability of winning. In fact, this is the basis for calculating probabilities in a binomial distribution.

The calculator above finds the number of possible combinations of things in a collection. You can also use a formula the calculate the number of combinations in a set.

### Combinations Formula

The following formula defines the number of possible combinations of *r* items in a collection of *n* total items.

C(n,r) = n!r!(n – r)!

Thus the number of possible combinations of *r* items in a set of *n* items is equal to *n* factorial divided by *r* factorial times *n* minus *r* factorial.

## N Choose K

Combinations are also sometimes referred to as *n choose k* where *n* is the total number of items and *k* is the number of items in the sample. Thus the number of combinations is equal to the number of ways you can choose *k* items from a set of *n* total items.

### N Choose K Formula

The *n choose k* formula is nearly identical to the combinations formula above, but the number of items in the sample is denoted *k*, and the total number of objects is denoted *n*.

C(n,k) = n!k!(n – k)!

## Combinations vs. Permutations

Combinations are very similar to permutations with one key difference: the number of permutations is the number of ways to choose *r* objects in a set of *n* objects in a unique order. Thus, with permutations, the order of the objects in the set is important.

In most cases, there will be more possible permutations of objects in a set because each different order of the items is considered a different permutation, but with combinations, the order is irrelevant, and those are considered the same combination of items.

**For example,** see the combinations and permutations of 2 letters in a set of the letters {A, B, C}.

**Combinations (3)**

(A, B)

(A, C)

(B, C)

**Permutations (6)**

(A, B)

(B, A)

(A, C)

(C, A)

(B, C)

(C, B)

Thus, for a sample of 2 objects in a total number of 3 objects, there are 3 combinations and 6 permutations.

### Permutations Formula

The formula below states how to calculate permutations:

P(n,r) = n!(n – r)!

Thus the number of possible permutations of *r* items in a set of *n* items is equal to *n* factorial divided by *n* minus *r* factorial.